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Peculiar velocities in redshift space: formalism, N-body simulations and perturbation theory

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J

ournal of Cosmology and Astroparticle Physics

An IOP and SISSA journal

Teppei Okumura,a,b Uroš Seljak,a,c,d Zvonimir Vlahd andVincent Desjacquesea Institute for the Early Universe, Ewha Womans University,

Seoul, 120-750 S. Korea
b Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),The University of Tokyo, Chiba, 277-8582 Japanc Department of Physics, Department of Astronomy, andLawrence Berkeley National Laboratory, University of California,
Berkeley, California, 94720 U.S.A.
d Physik Institut, University of Zurich,
Zurich, 8057 Switzerlande Département de Physique Théorique and Center for Astroparticle Physics (CAP),
Université de Genève, Genève, 1211 Switzerland
E-mail: teppei.okumura@ipmu.jp, useljak@berkeley.edu, zvlah@physik.uzh.ch,Vincent.Desjacques@unige.ch
Received December 17, 2013
Revised March 29, 2014
Accepted April 3, 2014
Published May 6, 2014

Abstract. Direct measurements of peculiar velocities of galaxies and clusters of galaxies
can in principle provide explicit information on the three dimensional mass distribution, but
this information is modulated by the fact that velocity field is sampled at galaxy positions,and is thus probing galaxy momentum. We derive expressions for the cross power spectrum
between the density and momentum field and the auto spectrum of the momentum fieldin redshift space, by extending the distribution function method to these statistics. The
resulting momentum cross and auto power spectra in redshift space are expressed as infinite
sums over velocity moment correlators in real space, as is the case for the density power
spectrum in redshift space. We compute each correlator using Eulerian perturbation theory(PT) and halo biasing model and compare the resulting redshift-space velocity statistics to
those measured from N -body simulations for both dark matter and halos. We find that inredshift space linear theory predictions for the density-momentum cross power spectrum aswell as for the momentum auto spectrum fail to predict the N -body results at very largec 2014 IOP Publishing Ltd and Sissa Medialab srl

doi:10.1088/1475-7516/2014/05/003

JCAP05(2014)003

Peculiar velocities in redshift space:formalism, N-body simulations andperturbation theoryscales. On the other hand, our nonlinear PT prediction for these velocity statistics, together
with real-space power spectrum for dark matter from simulations, improves the accuracy
for both dark matter and halos. We also present the same analysis in configuration space,computing the redshift-space pairwise mean infall velocities and velocity correlation function
and compare to nonlinear PT.

Keywords: galaxy clustering, power spectrum, redshift surveys, cosmological perturbation
theoryArXiv ePrint: 1312.4214

JCAP05(2014)003

Contents
1

2 Velocity moments power spectra in real and redshift space
2.1 Momentum power spectra in real space
2.2 Density power spectrum from the distribution function approach
2.3 Redshift-space distortions of momentum fields
2.4 Momentum power spectra in redshift space

3
3
4
5
6

3 N -body simulations

7

4 Fourier-space analysis
4.1 Dark matter in real space and in redshift space
4.1.1 Cross power spectrum between density and momentum
4.1.2 Auto power spectrum of momentum
4.2 Halos in redshift space
4.2.1 Cross power spectrum between density and momentum
4.2.2 Auto power spectrum of momentum

8
9
9
11
13
13
14

5 Configuration-space analysis
5.1 Mean pairwise infall momentum
5.1.1 Dark matter
5.1.2 Halos
5.2 Momentum correlation function
5.2.1 Dark matter
5.2.2 Halos

18
19
21
22
25
26
28

6 Effects of satellite galaxies on velocity statistics
6.1 Fourier space
6.2 Configuration space

29
31
32

7 Conclusions

34

A Real-space velocity statistics of halos
A.1 Fourier space
A.2 Configuration space

35
35
36

B Density-weighted vs pair-weighted velocities
B.1 Mean pairwise velocity
B.2 Velocity correlation function

38
38
40

1

Introduction

The peculiar velocity field probed by galaxies and clusters of galaxies is cosmologically very
important because it provides information on the three dimensional mass density fluctuations.
There are two observational ways to extract the information. The first one is a galaxy–1–

JCAP05(2014)003

1 Introduction

–2–

JCAP05(2014)003

redshift survey, where peculiar velocities induce redshift space distortions along the line ofsight (RSD) [1–4]. The information on peculiar velocities is entangled in the clustering throuh
the dependence on the line of sight angle and, thus, can be separated out from the intrinsic
clustering of galaxies. The second is a direct map of peculiar velocities [5–7], which canbe obtained by measuring the intrinsic distances to galaxies (e.g. by using Tully-Fisher or
Faber-Jackson relations) and comparing them to the galaxy redshift. Unlike redshift surveys,
peculiar velocity surveys furnish in principle a direct measurement of peculiar velocities.
However, the error on the velocity measurement is proportional to the distance, so the analysis
is limited to the nearby universe (z ∼ 0). The two types of surveys are largely complementary
to each other, and combining these two probes can tighten constraints on the growth rate ofcosmic structure and, therefore, on theories of modified gravity [8]. Observational analyses in
peculiar velocity surveys have been performed by [e.g., 9–12] (see also [13] for recent work).
More recently, it has been suggested that peculiar velocity measurements can be extended tohigher redshifts using Type Ia supernovae [14–16] or the kinetic Sunyaev-Zeldovich (kSZ, [17])effect [18]. The kSZ effect has been detected by [19] using high-resolution microwave sky maps
and a large galaxy redshift survey.
Theoretical studies have attempted to predict peculiar velocity statistics using linear
theory [5, 6, 20–23], halo models [18, 24], nonlinear PT [25–28] or empirical fitting functions [29]. All the above theoretical works were formulated in real space. In reality, what ismeasured in peculiar velocity surveys is the velocity field sampled in redshift space, and themeasured positions are distorted by RSD. Theoretically, the RSD effect on peculiar velocities
has been discussed previously in Fourier space statistics [30–33].
In this paper, we present a comprehensive analysis of the aforementioned issues for two
peculiar velocity statistics: the velocity-density correlation, which can be restated as the pairwise mean infall velocity, and the velocity-velocity correlation. We have developed and testeda phase space distribution function approach to RSD where the redshift-space density can be
written as a sum over mass or number weighted moments of radial velocity [34–38]. The corresponding RSD power spectrum can be written as a sum over auto and cross-correlators ofthese moments. By applying this technique to the momentum density field in redshift space,
we derive theoretical predictions for momentum power spectra in redshift space as a sum
over auto and cross correlators of density-weighted velocity moments. We present detailed
comparisons of these predictions computed using Eulerian nonlinear perturbation [37, 38] tothe measurements for dark matter and halos obtained from N -body simulations.
This paper is organized as follows. In section 2, we briefly review peculiar velocity
statistics in real space, and derive expressions in redshift space by extending the distributionfunction approach for the redshift-space density field to the redshift-space momentum field.
Section 3 describes the N -body simulations as well as our measurements of the Fourier space,peculiar velocity statistics considered in this paper. In section 4, we present the peculiar
velocity statistics in Fourier space, the density-momentum and momentum-momentum power
spectra measured from N -body simulations. We then make a detailed comparison with linear
and nonlinear perturbation theory. In section 5 we present the peculiar velocity statistics inconfiguration space, the mean pairwise velocity and the velocity-velocity correlation function.
We then discuss the configuration-space peculiar velocity statistics measured from N -bodysimulations and predicted using perturbation theory. In section 6, we test whether ournonlinear perturbation theory for halos can also model the peculiar velocity statistics ofsatellite galaxies. Section 7 is the conclusion of this paper. For a comparison to the redshiftspace velocity statistics we discuss in this paper, we present the corresponding real-spacestatistics in appendix A. In the analysis of velocity statistics in configuration space, statistics
using pair-weighted velocity is commonly used while we focus on the statistics using densityweighted velocity throughout this paper. In appendix B, we perform a detail comparison ofpair-weighted and density-weighted statistics and show that the latter are better defined andmore direct observables.

2

Velocity moments power spectra in real and redshift space

pk (x) = [1 + δ(x)] vk (x),(2.1)

where pk = p · ẑ and ẑ is the unit vector pointing along the line of sight. Equation (2.1) is
the L = 1 case of the moments of mass-weighted velocities TkL defined in [34] (see section 2.2
below).
2.1

Momentum power spectra in real space

By correlating the radial momentum density to the density field and to the radial momenr and second-order P r velocity
tum density itself, respectively, we obtain the first-order P01
11
statistics, asDEr
δ(k)|p∗k (k0 ) = (2π)3 P01(k)δ D (k − k0 ),
(2.2)
D
E
r
pk (k)|p∗k (k0 ) = (2π)3 P11(k)δ D (k − k0 ),(2.3)
where δ(k) and pk (k) are the Fourier transforms of δ(x) and pk (x), respectively. Quantities
with the superscript r mean that they are measured in real space, while later we will consider
these quantities in redshift space, denoted by the superscript s. The general definition ofr
PLL
0 was introduced in [34] and will be given in the next subsection.
The momentum field is related to the density field through the continuity equation, andin linear theory for biased objects it reads
r,hP00,lin
(k) =



−ikb
Hµf



r,h
P01,lin
(k) =



kb
Hµf

2

r,h
P11,lin
(k),

(2.4)

r is the ordinary power spectrum of a given density field, H(a) = aH(a), H(a) is
where P00
the Hubble parameter, µ is the angle between the Fourier mode and line-of-sight, b is the–3–

JCAP05(2014)003

In this section we briefly review theoretical formalisms for the velocity statistics in real
space, and also newly define the statistics in redshift space. Throughout this paper weadopt a plane-parallel approximation. In what follows the super/subscript m and h denotequantities for dark matter and halos, respectively. We will, however, omit the m and hwhen a given equation holds for both dark matter and halos, because much of our formalism
remains unchanged if the dark matter particles are replaced with biased tracers such as halos
or galaxies.
In peculiar velocity surveys, peculiar velocities of galaxies themselves are directly observable, but they are sampled at galaxy positions, so they are not volume-weighted but
density-weighted velocities. We can define the comoving momentum density field as p(x) =[1 + δ(x)] v(x), where δ = ρ/ρ̄ − 1 is an overdensity field and v is the comoving velocity field.
Since only the radial components of these vectors are observables, a more relevant relation isbias parameter and f = d ln D/d ln a with D the growth factor. The expression for dark
matter is obtained by substituting b = 1 in equation (2.4), thus the relations between powerr,hr,mspectra for dark matter and halos in linear theory are characterized by b as P00,lin= b2 P00,lin
,
r,h
r,m
r,h
r,m
P01,lin= bP01,lin
and P11,lin= P11,lin[34, 36].

2.2

Density power spectrum from the distribution function approach

L=0

TkL (k) is the Fourier transform of TkL (x), the moments of mass-weighted velocity, defined asZ
ma−3LTk (x) =d3 pf (x, p)vkL ,(2.6)
ρ̄
where f (x, p) is the phase space distribution function of particles at a phase-space position(x, p). For a single stream at x, equation (2.6) is simply written as TkL (x) = [1 + δ(x)] vkL (x)and it is obviously the L = 1 case if equal to equation (2.1). The equations above are applied
not only to dark matter but also to biased objects [36]. The power spectrum in redshift spacecan be expressed as an infinite sum of cross-power spectra between mass weighted velocity
moments in real space, as
0

0 
∞ X
∞
X
(−1)L ikµ L+L rs(2.7)
P00 (k) =PLL0 (k) ,L! L0 !
H
0
L=0 L =00r (k)(k − k0 ) = hT L (k)T ∗L (k0 )i. Note that we explicitly put the subscript 00
where (2π 3 )PLL0kks
to P to clarify that the so-cold redshift-space power spectrum is the auto correlation of thezeroth velocity moments in redshift space, since in the next subsection we discuss the power
spectra of velocity moments in redshift space.
By performing helicity decomposition, [34] determined the angular dependence of PLL0
and the redshift-space power spectrum was shown to be 2L X
2(L+L0 )4L∞ X∞
L  L+L0
X
X
X
1
k
(−1)
ik
(j)
(j)
s
j
P00
(k) =
P
(k)µ
+2
PLL0 (k)µj
LL
2
0
L! H
L! L ! H
L=0
j=2L
L=0L0 >Lj=(L+L0 ) or (L+L0+1)
X
s
j
≡
P00,µ
(2.8)
j (k)µ ,j=0,2,4,···(j)rswhere PLL0 are coefficients in expansion in powers of µj of contributions of PLL
0 to P00 . Thes2
coefficients P00,µj are the moments expanded in terms of µ , an alternative expansion to thecommonly-used Legendre multipole expansion,
Z 1
X
s
s
s
s
P00 (k) =P00,l (k)Pl (µ),
P00,l (k) = (2l + 1)
P00(k)Pl (µ)dµ.
(2.9)
0l=0,2,4,···–4–

JCAP05(2014)003

A phase-space distribution function approach was recently developed by [34, 39] for the
purpose of modeling the power spectrum of the density fields in redshift-space. The approach
was tested against dark matter in [35, 37] and extended to biased objects such as dark matter
halos and galaxies in [36, 38]. In this subsection we briefly review the formalism.
The redshift-space density field in Fourier space is described as a summation of momentsof density-weighted velocity in real space,
X 1  ikk Lsδ (k) =
(2.5)
TkL (k).
L! H

s is given by the velocity
We already know that the redshift-space power spectrum P00
statistics above as well as the density-density spectrum P00 in real space under the nonlinear
Kaiser approximation [3, 25, 36],
s,h
r,h
P00
(k) = P00(k) + 2f µ2



−ik
Hµf



r,h
P01
(k) + f 2 µ4



k
Hµf

2

r,h
P11
(k).

(2.10)

If we additionally assume standard linear theory in equation (2.10), we obtain the original
linear Kaiser formula, as

2.3(2.11)

Redshift-space distortions of momentum fields

A direct observable in a peculiar velocity survey are in fact peculiar velocities in redshift
space, which are affected by redshift-space distortions. The position of a particle in redshift
space s is distorted by its peculiar velocity along the line of sight as s = x + ẑvk /H. Thus
we can define the redshift-space momentum field as
psk (s) = [1 + δ s (s)] vks (s).

(2.12)

By extending the phase-space distribution function approach to the redshift-space momentum
field, the momentum density along the line of sight in redshift space is given byZZvk  ma−3
pk
pk 
vk 
ma−3
d3 p d3 x
f (x, p) δ D s − x − ẑ=d3 p
f s − ẑ , p .
psk (s) =
ρ̄
ma
H
ρ̄
ma
H
(2.13)
By Fourier transforming equation (2.13), we get−3 Z

pk
f (x, p) ei(k·x+kk vk /H)
ρ̄
ma
Z
Z
pk
ma−3
d3 x eik·x
d3 p
f (x, p)eikk vk /H .
=
ρ̄
ma

ma
psk (k) =d3 x d3 p(2.14)

Expanding the second integral in equation (2.14) as a Taylor series in kk vk /H, we get
ma−3
ρ̄

Z

pk
ma−3
d p
f (x, p)eikk vk /H =
ma
ρ̄

Z

X 1



3

=

L=0

d3 q f (x, p)

X 1
L=0

L!

ikk
H

L

L!

ikk /HL

TkL+1 (x).

vkL+1
(2.15)The Fourier component of the momentum fluctuation in redshift space is
psk (k) =X 1  ikk L
L=0

L!

H

TkL+1 (k) =

X
L=1

1
(L − 1)!



ikk
H

L−1

TkL (k).

(2.16)

This is the expression for the momentum density in redshift space corresponding to that forthe mass density (equation (2.5)).

–5–

JCAP05(2014)003

2 h2 r,ms,hP00,lin(k) = 1 + f µ2 /b P00,lin(k) = b + f µ2 P00,lin
(k).

2.4

Momentum power spectra in redshift space

As is the case with real space (equations (2.2) and (2.3)), the redshift-space overdensitymomentum and momentum-momentum power spectrum are defined as,1
E
D
0
s
(k
)
= (2π)3 P01(k)δ D (k − k0 ),δ s (k)|ps∗
k
D
E
0
3 s
D
0
psk (k)|ps∗(2.17)k (k ) = (2π) P11 (k)δ (k − k ).
Equations (2.5) and (2.16) give,
X X (−1)L0  ikk L+L0r
PLL
0 +1 (k)H0
L=0 L =0
L+L00 
H X X (−1)L ikkr=i
L0 PLL
0 (k),kk
L!L0 !
H
0
L=0 L =1
X X (−1)L0  ikk L+L0sr
P11 (k) =PL+1L
0 +1 (k)0!
L!LH
L=0 L0 =0L+L0 2 X X
0 (−1)L ikkHr=
LL0 PLL
0 (k).
0!
kk
L!L
H
0

L!L0 !

(2.18)

L=1 L =1

r (k) with L ≤ L0 need to beThus only the terms PLL00considered, each of which comes with a factor of 2 if L 6= L and 1 if L = L0 . If we introduceµ = kk /k = cos θ, we can write,
X (−1)L  ikµ L
X 1  kµ 2Lkµ srr−i P01 (k) =L P0L (k) +
L PLL
(k)
H
L!
H
(L!)2 H
L=1
L=1
X X (−1)L0  ikµ L+L0
r(2.19)
L0 PLL+2Re
0 (k),
0!
L!LH
L=1 L0 >L 2
X 1  kµ 2L
kµ
s
r
P11
(k) =
L2 PLL(k)
H(L!)2 H
L=1
X X (−1)L0  ikµ L+L0
r(2.20)+2Re
LL0 PLL
0 (k).
L!L0 !
H0
r (k) = P r (k)∗ .
As noted in [34] PLL
0L0 L

L=1 L >Ls in equation (2.8), we have
Considering the helicity decomposition as we did for P00
X
X
s
s
j
s
s
j
P01
(k) =
P01,µ
P11
(k) =
P11,µ
j (k)µ ,j (k)µ .
j=1,3,···(2.21)

j=2,4,···On large scales where linear theory can be applied, peculiar velocity statistics in redshifts,hs,hspace, P01
and P11, can be expressed as, 2kµ s,hkµ r,hkµr,hr,m−i P01 (k) = −i P01 (k) +P11(k) = f µ2 (b + f µ2 )P00,lin(k), (2.22)
H
H
H
 2
 2
kµ
kµ
s,h
r,h
r,m
P11 (k) =P11(k) = µ4 f 2 P00,lin(k).
(2.23)
HH
1sss
Note that unlike the definition here, we used the definition of PLL
0 (or PLL0 ) to describe contributions ofs
PLL0 in real space to the redshift-space power spectrum P (k) in our series of papers on RSD [34–37].

–6–

JCAP05(2014)003

s
P01
(k) =

We thus see that even in linear theory, the density-velocity correlation is not just proportional
to µ, but has an additional b + f µ2 dependence that arises from RSD. This has usually been
ignored in previous analyses [e.g., 19]. The expression of equation (2.22) is the same as that
presented by [30] who considered the cross correlation of redshift-space density and real-spacevelocity in the linear regime, because the velocity field in redshift space is equivalent to thatin real space for the linear theory limit (see equations (2.16) and (2.23), and see also [33]).

3

N -body simulations

r (k) and P s (k) are measured byredshift space Tks,L (k). Then the power spectrum PLL0
LL0
multiplying the modes of the two fields (or squaring in case of auto-correlation) and averaging
over the modes within a bin. The momentum field is normalized by H(a) = aH(a), thus
the momentum power spectra P01 and P11 respectively have the unit of [ h−1 Mpc]4 and[ h−1 Mpc]5 . To show the error of the mean for measured statistics, the scatter among
realizations needs to be divided by the square root of the number of the realizations. Three
measurements along different line-of-sight in each of 12 realizations are, however, not fully√
independent. To be conservative, we thus present the measured dispersion divided by 12
as the error of the mean in the following analysis.

–7–

JCAP05(2014)003

As in our previous works [35, 36], we use a series of N -body simulations of the ΛCDM cosmology seeded with Gaussian initial conditions [40]. The primordial density field is generatedusing the matter transfer function by CAMB [41]. We adopt the standard ΛCDM modelwith Ωm = 1 − ΩΛ = 0.279, Ωb = 0.0462, h = 0.7, ns = 0.96, σ8 = 0.807. We employ
10243 particles of mass mp = 2.95 × 1011 h−1 M in a cubic box of side 1600 h−1 Mpc. The
positions and velocities of all the dark matter particles are output at z = 0, 0.509, 0.989,
and 2.070, which are quoted as z = 0, 0.5, 1, and 2 in what follows for simplicity. We use 12
independent realizations in order to reduce the statistical scatters.
Dark matter halos are identified using the friends-of-friends algorithm with a linking
length equal to 0.17 times the mean particle separation. We use all the halos with equal toor more than 20 particles. In order to investigate the halo mass dependences of the velocity
statistics, each dark matter halo catalog is divided into subsamples according to the halo
mass. The subsamples are labelled as bin1, bin2, · · · , from the least massive halos. Since
the number density of halos is smaller at higher redshifts, we construct 4 halo subsamples
at z = 0 and 0.5, and 3 subsamples at z = 1. We do not consider the analysis of halos
at z = 2. The values of the linear bias parameter b1 determined using χ2 statistics for
r,mm
r,mh
(k) at 0.01 ≤ k ≤ 0.04 h Mpc−1 are b1 = 1.17, 1.45, 2.02 and 3.03(k)/P00b(k) = P00
at z = 0, b1 = 1.64, 2.16, 3.11 and 4.85 at z = 0.5, and b1 = 2.31, 3.17 and 4.64 at z = 1.
mh is the cross power spectrum of matter and halo density fields. See table 1 of [36] for the
P00
detail of the halo catalogs.
We measure the power spectra of velocity moments for dark matter and halos from oursimulation samples following [35, 36]. We assign the number-weighted velocity moments inreal space and in redshift space on 10243 grids using a cloud-in-cell interpolation method
according to the positions of particles. In measuring the velocity field in redshift space, we
distort the positions of particles along the line-of-sight according to their peculiar velocities
before we assign them to the grid. We regard each direction along the three axes of simulation
boxes as the line of sight and the statistics are averaged over the three projections of all
realizations for a total of 36 samples. We use a fast Fourier transform to measure theFourier modes of the number-weighted velocity moment fields in real space TkL (k) and in

4Fourier-space analysis

The purpose of this paper is to present the momentum cross and auto power spectrum fordark matter and halos in redshift space measured from N -body simulations and predict themusing linear theory and one-loop nonlinear perturbation theory. As we did in [37, 38], wes and P s , up to
consider terms that contribute to the redshift-space velocity statistics, P01
11r0
one loop in Eulerian PT, so we have PLL0 up to L + L ≤ 4 ands as µ2 andAmong these terms in equation (4.1), all the terms that contribute to −i(kµ/H)P01
4rµ , except for P04 , can be formulated using one-loop PT for dark matter [37] and halos [38].
r is of two-loop order, but we take into account the PT modeling
The µ4 term in (kµ/H)4 P04
of it because, as shown in [37], it has a non-negligible contribution. We thus include all thes in the following analysis, i.e.,terms that have the angular dependence of µ and µ3 to P01
24sµ and µ to −i(kµ/H)P01 . Likewise, we take into account all the terms that contributes . All the linear theory predictions presented below are computed using
to µ0 and µ2 in P11
CAMB [41].
r,mAs discussed in [37], modeling the matter power spectrum in real space, P00, plays anr,hr,mimportant role to predict both PLL0 and PLL0 . The matter power spectrum in real space is themost fundamental spectrum and there are many attempts to model it based on perturbation
theory for quasi-nonlinear scales [e.g., 42–46] and on calibration using N -body simulations
for fully nonlinear scales [e.g., 47–49]. In this paper, rather than modeling it we assume thatr,mr,mis equal to that, namely, the modeled P00
we have a perfect theoretical model for P00
measured from the N -body simulation. With this assumption we also have perfect modelr,mbecause the scalar part of the momentum field is related to the time derivativefor P01of the matter density field (see equation (3.7) of [34]). Note that we still use nonlinearr,m
PT for the vector part of P11
that was shown to be in a good agreement with N -bodyr,mr,mand the scalar part ofresult [37]. In the following analysis, thus, we use values of P00
, P01
r,m
P11 measured from N -body simulations as our prediction and use nonlinear PT at one-loopr,mr,morder for the other PLL
0 ’s and at two-loop for P04 . In this way we can investigate theaccuracy of nonlinear PT on prediction of nonlinear RSD, since all the possible deviations
from the N -body measurement come from the imperfect modeling of nonlinear RSD, as wellas the imperfect modeling of nonlinear bias in the case of analysis of dark matter halos.
r ’s in equations (4.1) and (4.2), except P r
As we have seen in [36, 37], all the PLL
001rand P11 , have terms proportional to velocity dispersion σv2 . Briefly, at one loop order, ther contains a term proportional to σ 2 P r , the correlators P r and P risotropic part of P02
v 00
12
03
r , P r is simply proportional to σ 2 P r , and P r contains
contain a term proportional to σv2 P01
v 01
13
22
r and another proportional to σ 4 P r . The correlator P r contains
a term proportional to σv2 P02
v 00
04
r and another proportional to σ 4 P r at two loop order. Usinga term proportional to σv2 P02
v 00

–8–

JCAP05(2014)003

 
 2
 2
kµ s
kµ r
kµ
kµ
i kµ 3 rr
r−i P01 (k) = −i P01 (k) +P11 (k) −P02 (k) +P03 (k)
H
H
H
H
2 H
 
 4
 
 3
1 kµ 4 rkµ
1 kµ 4 r
kµrr
P12 (k) +P04 (k) −P13 (k) +P22 (k),
− 2i
H
6 H
H
2 H
(4.1)
 2
 2
 3
 4
 4
kµ
kµ
kµ
kµ
kµ
s
r
r
r
r
P11
(k) =
P11
(k) − 2i
P12
(k) −
P13
(k) +
P22
(k).
H
H
H
H
H
(4.2)

perturbation theory, the velocity dispersion is evaluated asZ1
d3 q Pθθ (q)
2
σv =
,
3
(2π)3 q 2

(4.3)

4.1

Dark matter in real space and in redshift space

In figure 1 we show the density-momentum cross power spectra and the momentum auto
power spectra measured for dark matter and halos as functions of (k, µ) both in real space andredshift space at z = 0. We adopt the constant µ binning into five bins between 0 ≤ µ ≤ 1,
but only four µ bins among the five are plotted, 0 ≤ µ ≤ 0.2 (green), 0.2 ≤ µ ≤ 0.4 (yellow),
0.4 ≤ µ ≤ 0.6 (red) and 0.8 ≤ µ ≤ 1 (blue). The corresponding linear theory predictions areshown as the dotted line with the same color. Our nonlinear PT predictions are shown as
the solid lines. In the rest of this section we will see the results in detail. First let us focus
on the results for dark matter that are shown in the left column of figure 1, while those forhalos will be presented in the next subsection.
4.1.1

Cross power spectrum between density and momentum

At the top of the left column of figure 1 we show the density-momentum power spectrum
r,m(k, µ), multiplied by k 2 for dark matter. This is the quantity that has
in real space P01
been already computed and presented in [35, 37], and we show it here to compare with thecorresponding redshift-space spectrum. As we have seen in [35, 37], this statistics approaches
linear theory at large scales shown as the dotted lines, while for k > 0.1 h Mpc−1 it exceeds
r,mis used as our theoretical prediction,
linear theory. Because the N -body measurement of P01
r,mby construction.
the prediction shown as the solid lines coincides with the measured P01
Next, at the left panel of the second row of figure 1 the density-momentum powers,mspectrum for dark matter measured in redshift space, P01(k, µ), is shown. There are two
r,m
r,m
s,m
s,m
terms, P01 and P11 , that have linear-order contributions to P01. As expected, P01
r,mstarts to deviate from linear theory at larger scales than P01 . At k ' 0.15 h Mpc−1 thes,msign of P01
is changed, because of the FoG effect, mainly from P02 . The more detailed
comparison is presented for four redshifts at the top row of figure 2 where we plot the ratios,ms,mof P01to the corresponding linear power spectrum without BAO wiggles P01,no-wiggle
=
r,m
r,m
2
(1 + f µ )P01,no-wiggle . The linear theory no-wiggle power spectrum in real space P01,no-wiggles,mis computed using the formula of [50]. Here P01,no-wiggle
has two terms, one proportional to
s,m
s,m
3
µ and another proportional to µ , thus the ratio P01 /P01,no-wiggle
has angular dependence.
−1
One can see that except at very large scales k ∼ 0.01 h Mpc , linear theory cannot explain
the N -body results at all. On the other hand, our nonlinear PT prediction agrees well with
the N -body results. Our PT calculation also well explains the trend of the change of signss,mof P01. However, at higher µ, the deviation of the PT prediction from the N -body result
starts at larger scales: at µ = 0.9 k ∼ 0.05 h Mpc−1 for z = 0 and k ∼ 0.08 h Mpc−1–9–

JCAP05(2014)003

where Pθθ is the velocity divergence power spectrum. While linear theory gives σv '6f D h−1 Mpc, which is a good approximation for halo velocities [38], this does not properly
take account of small scale contributions which come from within virialized halos where PT
cannot be applied. In order to take into account the nonlinear contributions to the velocity
r,m
2
2
2
2
2
dispersion, we add the corrections σLL
0 to σv in PLL0 as σv → σv +σLL0 . Rather than treating
the nonlinear correction σLL0 as free parameter, we simply adopt the best fit parameters ofr,mσLL0 , which were obtained for the real-space PLL
0 and shown in table 1 of [37].

k2Pr01(k,µ)

Density - momentum in real space

102

Halos (bin1)

Dark matter
101

Halos (bin3)

Density - momentum in redshift space

101

Momentum - momentum in real space

k3Pr11(k,µ)

100
3
10

102

101

sim

µ=0.1
0.3
0.5
0.9

model

µ=0.1
0.3
0.5
0.9

linearMomentum - momentum in redshift space

k3Ps11(k,µ)

100
103

µ=0.1
0.3
0.5
0.9

102

101

100

0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 1. Density-momentum cross power spectra in real space P01 (k, µ) (first row ) and in redshiftsspace P01
(k, µ) (second row ), and momentum auto power spectra in real space P11 (k, µ) (third row )sand in redshift space P11(k, µ) (bottom row ), all measured at z = 0. From left to right we show theresults for dark matter, halos with mass bin 1, and halos with mass bin 3. P01 and P11 are respectively
multiplied by k 2 and k 3 for clarity. The points with different color show the results with different µ
values, quoted in the third row, with the width of ±0.1. The sold lines show our PT predictions andthe negative values are shown as the dashed lines. The corresponding linear theory predictions areshown as the dotted line with the same color.
s has the angular dependence of µ2j (j =for z = 2. This is expected because −(ikµ/H)P01
1, 2, · · · ), while we used terms up to µ4 using PT, and the neglected higher order terms are
more significant for higher µ. On the other hand, such higher-order terms are significantlys,msuppressed for smaller µ, thus our one-loop PT agrees with N -body results of P01
at µ = 0.3
up to much smaller scales. For comparison, the density-momentum spectrum in real space
r,m
r,m
P01
(k, µ)/P01,no-wiggle(k, µ) is shown in the top row of figure 2 as the black dashed liner has angular dependence of P r (k) = µP r
(angle independent). Since P01
01
01,µ1 (k), the ratio ofr,m
the measured P01 to the corresponding no-wiggle linear spectrum becomes isotropic. The– 10 –

JCAP05(2014)003

k2Ps01(k,µ)

102

s,m
Ps,m
01 (k,µ)/P01,no-wiggle(k,µ)

dark matter
z=0

z=0.5

z=1

z=2

1
0.8
real space

0.6

redshift space
sim model
µ=0.3
0.5
0.9

0.4
0.2

linear

1.5

1

0.5

µ=0.3
0.5
0.9

0

sim model

linear

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3
k [h Mpc-1]

k [h Mpc-1]

k [h Mpc-1]

k [h Mpc-1]

Figure 2. Top row : density-momentum cross power spectra for dark matter in redshift space divideds,ms,mby the no-wiggle spectra [50], P01/P01,no-wiggle. From the left to right, results are shown for z = 0,0.5, 1 and 2. The solid lines show our PT predictions and the negative values are shown as the dashedr,mr,mlines. For comparison, the results in real space, P01/P01,no-wiggle, are shown as the black dashed line
(it is angle independent). The corresponding linear theory prediction is isotropic and equivalent in real
r,m
r,m
s,m
s,m
space and redshift space, P01,lin/P01,no-wiggle= P01,lin/P01,no-wiggle, shown as the dotted lines. Bottoms,mr,mrow : ratios of the spectra in redshift space and real space, P01(k, µ)/P01(k, µ). The corresponding
nonlinear PT prediction is shown as the solid (positive) and dashed (negative) lines with the samecolor as the measurement, while linear theory prediction is as the dotted lines.
s,mr,mratios of the cross power spectrum in redshift space to that in real space, P01
/P01
, are
shown in the bottom panels of figure 2. At the linear theory limit this ratio gives (1 + f µ2 ),shown as the dotted horizontal lines. Nonlinear PT can predict the numerical results forµ = 0.9 up to k ' 0.05 h Mpc−1 at z = 0 and to k ' 0.08 h Mpc−1 at z = 2, and to thesmallest scales probed here for µ = 0.3. Note that we use µ = 0 simulation results as thestarting point.

4.1.2

Auto power spectrum of momentum

Third, we show at the left panels of the third and fourth rows in figure 1 the momentum autos,mr,mand in redshift space P11, respectively. In linear theory
power spectrum in real space P11
the two statistics are equivalent (equation (2.23)) and the behaviors of these measurementsr,mare indeed similar. As we have seen in [35, 37], P11
exceeds linear theory at small scalesr,mjust like P01 but the nonlinear effect is stronger. For prediction of the auto momentum
spectrum in real space, we use the measurement from N -body simulation for the scalar partr,mand nonlinear PT for the vector part. Because the vector part of P11
is well modeled usingr,mone-loop nonlinear PT [37], the measured P11 perfectly agrees with our prediction. Unlike
s,m
s,m
r,m
P01
, P11
does not change the sign at all scales because there is no contribution from P02
that has a large FoG effect. The momentum auto power spectra of dark matter in real space
and in redshift space divided by their corresponding no-wiggle spectra are shown at the firsts,mrow in figure 3. Interestingly, the momentum auto power spectrum in redshift space P11
is better predicted using our nonlinear PT than the density-momentum cross spectrum in– 11 –

JCAP05(2014)003

r,m
Ps,m
01 (k,µ)/P01 (k,µ)

1.2

s,m
Ps,m
11 (k,µ)/P11,no-wiggle(k,µ)

redshift spaceµ model
0.1
0.3
0.5
0.9

dark
matter10

redshift space
µ sim
0.1
0.3
0.5
0.9

1
z=0

z=0.5

z=1

1.4

z=2
µ=0.1
0.3
0.5
0.9

1.2

sim model

10.8
linear
0.6
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3
k [h Mpc-1]

k [h Mpc-1]

k [h Mpc-1]

k [h Mpc-1]

Figure 3. Top row : momentum auto power spectra for dark matter in redshift space divided by thes,ms,mno-wiggle spectra P11/P11,no-wiggle. From the left results are shown for z = 0, 0.5, 1 and 2. The solidlines show nonlinear PT predictions with the same color as the measurement, while the linear theoryr,mr,mprediction is shown as the black dotted line. For comparison the real-space result P11/P11,no-wiggle
is shown as the dashed lines. Bottom row : ratios of the spectra in redshift space and real space,
s,m
r,m
P11
(k, µ)/P11(k, µ). The solid lines and the horizontal dotted lines are the nonlinear and linear
theory predictions, respectively.
s,ms,mredshift space P01. As is the case with P01, however, the deviation of our PT prediction
from the simulation result is larger at higher µ, since we use only the terms up to µ4 fors while it has the infinite angular dependence of µ2j (j = 1, 2, · · · ). The bottom(kµ/H)2 P11
row of figure 3 shows the ratio of the momentum auto spectra in redshift space and in realr,ms,m, that is just unity in linear theory. Although the effect of RSD on P11 is/P11space, P11
a fully nonlinear effect, that changes the amplitude of P11 by ∼ 50% at k ' 0.1 h Mpc−1
at z = 0 and ∼ 15% even at z = 2. These trends are well captured by our one-loop PT
calculations.
Let us consider the cross-correlation coefficient between density and momentum in redshift space, defined as,
P s,m (k)Rs,m (k) = p s,m01
.
(4.4)
s,m
P00 (k)P11(k)

In linear theory the dark matter density field is perfectly correlated with the momentum field
thus the cross-correlation coefficient is just unity. We show in figure 4 the cross-correlation
coefficient for dark matter in redshift space as functions of (k, µ). For comparison we also show
the cross-correlation coefficient defined in real space, Rr,m , that is a similar quantity presented
by [51] for the velocity field instead of the momentum field. Because the N -body measurement
and prediction of Rr,m are almost identical, we show only the latter as the dashed lines. Ther,mangular dependence of Rr,m comes from the nonlinear, isotropic part of P11. If only the
r,m
r,m
r,m
scalar part of P11 is used, the coefficient becomes isotropic R (k) = R (k) [36]. Becauseof the decoherence of density and momentum fields, Rr,m approaches 0 at small scales. As
for the cross-correlation coefficient in redshift space, Rs,m (k, µ), since the momentum field– 12 –

JCAP05(2014)003

r,m
Ps,m
11 (k,µ)/P11 (k,µ)µ real space
0.1
0.3
0.5
linear
0.9

1
z=0

z=0.5

z=1

z=2

Rs,m(k,µ)0.5

redshift space
sim model

0

µ=0.1
0.3
0.5
0.7
0.9

-0.5
0.2

0.3

0.1

k [h Mpc-1]

0.2

0.3

k [h Mpc-1]

0.1

0.2

0.3

k [h Mpc-1]

0.1

0.2

0.3

k [h Mpc-1]

Figure 4. Cross-correlation coefficient of the density and momentum for dark matter at four redshifts
in redshift space Rs,m (k, µ). The sold lines show our PT predictions. In linear theory the crosscorrelation coefficients are simply unity, as denoted by the black dotted lines. For comparison thecoefficient in real space is shown as the dashed lines.

here is correlated with the redshift-space density field that is distorted by the momentum
field, it does not approach 0 but goes to negative values. The trend is well captured by ournonlinear PT prediction. However, at low redshifts the deviation from N -body results getslarger and larger for higher k.
4.2

Halos in redshift space

Let us move onto results obtained for biased objects, dark matter halos. As mentioned above,
halo velocity dispersion is well predicted by linear theory with weak dependence of halo mass,so we use the linear theory prediction, σv /f D = 6 h−1 Mpc. There is an additional issue toh , that is halo biasing. Following [38] we adopt an Eulerian biasing model [52]model halo PLL0δh [δm ] = b1 δm (x) + bs 
 b3 3b2  22i +δm (x) − hδms(x)2 − hs2 i + δm
(x),
2
2
6

(4.5)

where bs is the nonlocal bias term that comes from the tidal tensor. With bias renormalizationb3 can be absorbed into b1 , and bs can be related to b1 through bs = − 27 (b1 − 1) [53]. Thus
we will consider a set of the two bias parameters (b1 , b2 ) for each halo subsample. We use
h with the best fit values of the halo biasing (b , b ) found by [38]. As is thethe models of PLL0
1 2
case in section 4.1, we assume that we can model the dark matter power spectrum in realm , thus we use P r,m , P r,m and the scalar part of P r,m measured from simulations.
space P00
00
01
11
4.2.1

Cross power spectrum between density and momentum

At the first row in figure 1, the middle and right panels show the density-momentum crosspower spectra of the lightest halos (bin1) and the second most massive halos (bin3), rer,hr,hspectively, in real space P01
measured at z = 0. Just like the case for dark matter, P01
for halos exceeds linear theory at small scales. Because we studied the density-momentum
power spectrum of halos in real space in detail in [36, 38], the result is shown in appendix A
only for z = 0 as a comparison to the corresponding redshift-space results shown below.
s,hThe cross power spectrum for halos in redshift space P01(k, µ) is shown at the middle
and right panels of the second row in figure 1. The spectrum for halos is closer to linear
theory than that for dark matter. It is because the power spectrum in redshift space is– 13 –

JCAP05(2014)003

0.1

real
space

4.2.2

Auto power spectrum of momentums,hr,h. Theyand P11
Next we consider the auto-power spectrum of momentum for halos P11
have an isotropic term thus affected by shot noise [34, 36]. In our previous works it was notr,his canceled out by that coming from the densityimportant because the shot noise on P11r,henergy density spectrum P02 when their contributions to the redshift-space power spectrums,his considered. Here, because we consider the momentum spectrum itself, the shot noise
P00
needs to be estimated and subtracted. [54] defined the scale-dependent shot noise for the halodensity field σn,00 (k) as δh (k) = bδm (k) + σn,00 (k), then the shot noise for the halo density
spectrum was estimated as
r,m
r,mh
r,h
2
(k) + b21 P00(k),(k) − 2b1 P00
σn,00(k) = P00

(4.6)

r,mh
where P00
is the cross power spectrum of matter and halo density fields in real space. If
one assumes the Poisson model the contribution of the shot noise to the halo power spectrum
2is simply σn,00= 1/n, where n is the halo number density. With the analogy to the methodof [54] above, we can estimate the shot noise for the momentum spectrum as
r,h
r,mh
r,m
2
σn,11
(k) = P11(k) − 2P11(k) + P11
(k),

(4.7)

r,mh
where P11
is the real-space cross power spectrum between matter and halo momentum
fields. Here we assumed that the momentum bias is unity at the large scale limit, which isa reasonable assumption as is clear from [36]. Again, under the assumption of the Poisson2
2model we have σn,11= hvk2 i/n = hvk2 iσn,00[11, 30].

– 14 –

JCAP05(2014)003

expanded in terms of kµvk /H, where radial velocity vk is much smaller for halos than fordark matter and the suppression of the amplitude by the nonlinear velocity dispersion ispartially cancelled out with the nonlinear clustering as we have seen in RSD of halos [36].
s has a different factor, e.g., P r + P r in P s is twice
Because each term of the expansion of P01
11
02
01
s,h
r
s
as large relative to P01 as that in P00 , the halo cross spectrum in redshift space P01
changess,hs,hsign at smaller scales than P00 . The halo mass dependence of P01 is shown at the first rowss,mof figures 5, 6 and 7 for z = 0, 0.5 and 1, respectively. As is the case with dark matter P01,
N -body results for halos disagree with linear theory prediction except for the largest scalek ≤ 0.02 h Mpc−1 . Although the velocity dispersion of halos is so small that linear theorys,hcan be applied, the amplitude of P01
is suppressed relative to linear theory prediction for all
the halo subsamples for all the redshifts. Our nonlinear prediction at one-loop level with aset of bias (b1 , b2 ) significantly improves the accuracy. Except the lowest mass bin at z = 0,
the nonlinear PT well predicts the scale dependence up to small scales including the scales,hwhere the sign of P01
turns negative. The ratio of the cross power spectrum in redshift spaces,hr,hand in real space, P01 /P01
, is shown at the bottom rows in figures 5, 6 and 7. The mores,hh on linear scales becomes
massive halos we focus on, the more isotropic the ratio P01/P01
since the anisotropy induced by RSD is parameterized by f /b. Although the overall trendof each simulation result is well captured by nonlinear PT, a careful look at the comparison
reveals small differences between N -body results and the PT prediction. For example atz = 0.5, they agree with each other for mass bin1 up to k ∼ 0.15 h Mpc−1 for µ = 0.3 andup to k ∼ 0.13 h Mpc−1 for µ = 0.9, and for mass bin3 up to k ∼ 0.2 h Mpc−1 for µ = 0.3
s (k, µ) along the line of sight isand up to k ∼ 0.11 h Mpc−1 for µ = 0.9. The prediction of P01
less accurate than that perpendicular to the line of sight because we consider PT at one-loop
level thus the angular dependence is truncated at µ3 .

s,h
Ps,h
01 (k,µ)/P01,no-wiggle(k,µ)

Redshift space
bin1

bin2

bin3

1
0.8
0.6
0.4

1

0.5
µ=0.3
0.5
0.9

sim

µ=0.3
0.5
0.9

model

µ=0.3
0.5
0.9

linear

0
0.05

0.1

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3

k [h Mpc-1]

r,h
Ps,h
01 (k,µ)/P01 (k,µ)

s,h
Ps,h
01 (k,µ)/P01,no-wiggle(k,µ)

Figure 5. Top row : density-momentum cross power spectra for halos in redshift space divided bys,ms,mthe no-wiggle spectra at z = 0, P01/P01,no-wiggle. Bottom row : ratios of the spectra in redshift spaces,mr,mand real space, P01(k, µ)/P01(k, µ). The corresponding linear theory predictions are shown as thedotted line with the same color, while our nonlinear PT predictions are as the solid and dashed lines
for positive and negative values, respectively.

1.2

Redshift space
bin1

bin2

bin3

1
0.8
0.6
0.4

1

0.5
µ=0.3
0.5
0.9

sim

µ=0.3
0.5
0.9

model

µ=0.3
0.5
0.9

linear

0
0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15

0.2

0.25

0.3 0.05

k [h Mpc-1]

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 6. Same as figure 5 but for z = 0.5.

In the top panels of figure 8 we show the shot noise measured for the halo momentum
2 . The horizontal lines with the same color as the measured points are the correfield σn,11sponding Poisson model. The bottom panels show the deviation of the shot noise from the
2 (k) − hv 2 i/n. As is clear from the figure, the behavior of the shot noise
Poisson model, σn,11kmeasured from simulations differs among halo subsamples with different mass, and it changes– 15 –

JCAP05(2014)003

r,h
Ps,h
01 (k,µ)/P01 (k,µ)

1.2

s,h
Ps,h
01 (k,µ)/P01,no-wiggle(k,µ)

Redshift space
bin1

bin3

bin2

1
0.8
0.6
0.4

1

0.5
µ=0.3
0.5
0.9

sim

µ=0.3
0.5
0.9

model

µ=0.3
0.5
0.9

0
0.05

0.1

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

linear

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 7. Same as figure 5 but for z = 1.

z=0

bin4

z=0.5

bin3

z=1

6

10

bin3

σ2n,11(k)

bin4

bin2

105

bin2

bin3

gal
bin2

bin1

bin1
4

bin1

10
σ2n,11(k) – <v2//>n–1

105

104

103

0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 8. Top: shot noise power spectrum for momentum power spectrum σn,11 measured using thematter-halo cross power spectra. The horizontal lines with the same color are the shot noise predictedwith the Poisson model, hvk2 i/n. Bottom: the difference between the measurement and the Poissonmodel. The filled and open points show positive and negative values, respectively.

a lot as a function of scale. It is expected to some extent, because even though equation (4.6)
is the true expression for the non-Poissonian shot noise for the halo density spectrum, thatfor the halo momentum spectrum (equation (4.7)) is just the analogy and holds only in thelimited case where the momentum field is uncorrelated with stochasticity. Moreover, we haver,htested if adopting equation (4.7) improved the accuracy of modeling P11
and found that theresults were not improved very much compared to the case with the Poisson shot noise. For– 16 –

JCAP05(2014)003

r,h
Ps,h
01 (k,µ)/P01 (k,µ)

1.2

s,h
Ps,h
11 (k,µ)/P11,no-wiggle(k,µ)

bin2

bin3

10

µ=0.1
0.3
0.5
0.9

1

sim

µ=0.1
0.3
0.5
0.9

model
linear

1

0.8
0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 9. Top row : momentum auto power spectra for halos in redshift space divided by theirs,hs,hno-wiggle spectra at z = 0, P11/P11,no-wiggle. Bottom row : ratios of the spectra in redshift spaces,hr,hand real space, P11(k, µ)/P11(k, µ). The corresponding linear theory predictions are shown as thedotted line with the same color while our nonlinear PT predictions are as the solid lines.

simplicity, we thus assume in the following analysis the Poisson model to subtract the shot
s,h
r,h
.
and P11noise from the measured auto power spectrum of halo momentum fields P11r,hThe resulting auto power spectra of halos for mass bin1 and bin3 in real space P11
at z = 0 after the shot noise is subtracted are shown respectively at the middle and rightr,hexceeds linear theory at large
panels in the third row of figure 1. The amplitude of P11r,hr,his well explained by nonlinear PT. The behavior of P11
scales similarly to the case of P01
for halos with the lowest mass, while there are systematic deviations for the result with more
massive halos. It is most likely because the assumption of the Poisson model for shot noise
hvk2 i/n is incorrect when the number density n is small. Because exploring the shot noise on
the momentum power spectrum is not the main purpose of this paper, the accuracy of ourmodel for the real-space spectra of momentum field with different halo mass is presented inappendix A. The effect of shot noise on the momentum power spectrum needs to be studied
in detail when analyzing real surveys.
s,h(k, µ). The
Finally we present the auto power spectrum of halos in redshift space P11
resulting redshift-space spectra with the shot noise subtracted for mass bin1 and bin3 at
z = 0 are shown at the middle and right panels in the bottom row of figure 1. The nonlinear
PT prediction for halos in redshift space achieve almost the same precision compared to
the real-space results, and has better agreement to N -body results for halos than for dark
matter. The ratios of these redshift-space momentum spectra to the corresponding linear
no-wiggle spectra are presented at the second rows in figures 9, 10 and 11 for z = 0, 0.5, and1, respectively. Overall, our nonlinear PT at one-loop level for the momentum power spectra
s,h
s,h
r,h
of halos in redshift space P11
is as accurate as that in real space. The ratio P11/P11
is shown at the bottom rows of figures 9, 10 and 11. The ratio is closer to unity, linear
theory prediction, than that for dark matter. The deviation from unity which is caused byfully nonlinear RSD and expected to be small is still at ∼ 20% levels. In previous theoretical– 17 –

JCAP05(2014)003

r,h
Ps,h
11 (k,µ)/P11 (k,µ)

Redshift space
bin1

s,h
Ps,h
11 (k,µ)/P11,no-wiggle(k,µ)

bin2

bin3

10

µ=0.1
0.3
0.5
0.9

1

sim

µ=0.1
0.3
0.5
0.9

model
linear

1

0.8
0.05

0.1

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3

k [h Mpc-1]

r,h
Ps,h
11 (k,µ)/P11 (k,µ)

s,h
Ps,h
11 (k,µ)/P11,no-wiggle(k,µ)

Figure 10. Same as figure 9 but for z = 0.5.
Redshift space
bin1

bin2

bin3

10

µ=0.1
0.3
0.5
0.9

1

sim

µ=0.1
0.3
0.5
0.9

model
linear

1

0.8
0.05

0.1

0.15

0.2

0.25

0.3 0.05

k [h Mpc-1]

0.1

0.15

0.2

0.25

0.3 0.05

k [h Mpc-1]

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 11. Same as figure 9 but for z = 1.

studies this effect has not been studied, but it needs to be taken into account. Our theoreticalprediction is well able to predict the ∼ 20% level effect. In order to improve the predictions,hup to smaller scales, however, we need to model the shot noise on P11
more correctly than
the Poisson model.

5

Configuration-space analysis

Observationally, peculiar velocity statistics have been studied more intensively in configuration space than in Fourier space. In this section we study two basic velocity statistics– 18 –

JCAP05(2014)003

r,h
Ps,h
11 (k,µ)/P11 (k,µ)

Redshift spacebin1

in configuration space: the mean infall velocity and velocity auto correlation function, thatare respectively related to the density-momentum cross power spectrum and momentum
auto power spectrum in Fourier space which we have studied so far. We define the Fouriertransform of the power spectra of the density-weighted velocity moments as
D
E Z
X,L
X,L0
X
0
ξLL
(r)
+
δ
δ
=
T
(x
)T
(x
)
=
0
a
0L 0L
b k
k

d3 k XP 0 (k, µ)e−ik·r ,(2π)3 LL

(5.1)

5.1

Mean pairwise infall momentum

The mean streaming velocity between pairs of galaxies, vmean , was originally introducedby [5] in the context of the BBGKY theory. Because only the line-of-sight component ofvelocity, vk (x) = v(x) · ẑ, is measurable in observations, we consider the mean streaming
velocity along the line of sight. The density-weighted, line-of-sight pairwise velocity for bothdark matter and halos is described as [5, 20, 25];r
[1 + δ(xa )] [1 + δ(xb )] vk (xb ) − vk (xa ) = −2 δ(xb )pk (xa ) = −2ξ01
(r),

(5.2)

where pk (x) is the momentum density field given in equation (2.1). As in section 2, a quantity with superscripts r is defined in real space, distinguished from redshift-space quantities
denoted with s that we will discuss in the following, and the superscripts m and h are omitted
when a given equation holds both for dark matter and halos. Since the mean infall velocity
in configuration space is related to the density-momentum cross power spectrum throughr has only the dipole component, ξ r (r) = ξ r (r)µ. Note that
equations (5.1) and (5.2), ξ01
01
01,1
the mean pairwise velocity in equation (5.2) is velocity weighted by density, thus the mean
pairwise momentum. It is related to the commonly used, pair-weighted velocity throughrr = [1 + ξ r ]v r
−2ξ01
kmean , where vkmean is the radial component of the pair-weighted infall
velocity, vkmean = vmean · ẑ. We will make a comparison of the density-weighted velocity
statistics to pair-weighted velocity statistics in appendix B. In linear theory, the equation of
the real-space mean streaming is described for halos as [5, 20, 25, 55, 56];
Z
f
r,h
r,h
ξ01,lin(r) = µ 2
kdkP00,lin(k)j1 (kr).
(5.3)
2π b
The linear-theory expression for dark matter is obtained by replacing the superscript h withm and substituting b = 1. See [18, 56] for nonlinear prediction of the mean streaming velocity
based on the halo model and [27] for prediction based on Lagrangian perturbation theory [57].
In redshift space, where the radial velocity components can be measured in peculiar
velocity surveys, we can also write down the mean streaming momentum in the same way, as
D
h
iE
D
E
s
[1 + δ s (xa )] [1 + δ s (xb )] vks (xb ) − vks (xa ) = −2 δ s (xb )psk (xa ) = −2ξ01
(r).
(5.4)

– 19 –

JCAP05(2014)003

where r = xb − xa , TkL (x) is defined in equation (2.6), δLL0 is the Kronecker delta, andr and ξ s are
the superscript X is r for real pace and s for redshift space. The functions ξ00
00
the well-known density correlation functions for a given sample in real space and in redshift
X = ξ X . In this section we present the velocity
space, respectively, and simply written as ξ00
statistics in real space and redshift space, and relate them to the correlation functions of thevelocity moments defined above. We also derive the contribution from RSD to the Kaiser
limit for the mean infall velocity in redshift space.

Note that although the velocity terms in the r.h.s. of equations (5.2) and (5.4) look thesame, r = xb − xa in equation (5.4) is the separation measured in redshift space. Usings (x) asequations (2.21) and (5.1), we can derive full angular dependence of ξ01
X
s
s
j
ξ01
(r) =
ξ01,µ
(5.5)
j (r)µ .
j=1,3,···
s , we have additional anisotropic terms compared to
At the linear theory limit for ξ01

rξ01,lindue to linear RSD as follows:

where µlk = ẑ · k̂ and µkx = k̂ · x̂, and µlk = µkx since we assume the distant-observerssapproximation. There are thus ξ01,µ1 and ξ01,µ3 in equation (5.5) that have linear ordercontributions.
We measure the velocity statistics from N -body simulations using the direct pair counting in configuration space. Since only the line-of-sight component of velocity is observablein peculiar velocity surveys, an estimator for the mean pairwise velocity in real space using
only the observables was proposed by [58, 59], asPXXi,j (vi · x̂i − vj · x̂j )cijX
P 2vmean (r) =,(5.7)
i,j cij
where the superscript X = r or s andcij = r̂ ·x̂i + x̂j
.
2

(5.8)

cij is simply equal to µ under the distant observer approximation. This estimator was
adopted by several observational and numerical studies [e.g., 10, 18] and it is based on mean
pair-weighted velocity. Throughout this paper except in appendix B, we analyze the densityweighted velocity in this paper, thus we adopt the following estimator [e.g., 60],
P
X
X

 X
i,j (vi · x̂i − vj · x̂j )cij
X
X
P 2
− 2ξ01 (r) = 1 + ξ (r) vmean (r) =,(5.9)
i,j cij,rand−1 P 2P
where i,j c2ij,rand = 1 + ξ X (r)i,j cij , and cij,rand is the same as cij but without clustering. This estimator gives the dipole component of the mean pairwise momentum. As we
often do for the monopole component for the density
correlation function, we omit the orderR
X (r) ≡ ξ X (r) = 3 1 ξ X (r)P (µ)dµ. In real space it is simply
of l = 1 for the dipole, thus ξ01
1
01,1
0 01
3 s
r
r
s
s
s
ξ01 (r) = ξ01,µ1 , while in redshift space we have ξ01 (r) = ξ01,µ1 + 53 ξ01,µ3 + 7 ξ01,µ5 + · · · .

– 20 –

JCAP05(2014)003

DEs,hξ01,lin(r) = δ s,h (xb )vks,h (xa )ZZ
kdk r,m= ifP(k)dOk e−ik·r (b + f µ2lk )µkx
(2π)3 00,lin
Z
ZED
kdk r,m2
hhPdOk e−krµkr µ3lk
= δ (xb )vk (xa ) + if(2π)3 00,lin
ZZ
kdk r,m23r,h2−krµkr= ξ01,lin (r) + ifPdOk e
P3 (µlk ) + P1 (µlk )(2π)3 00,lin
5
5


 2 Z

3
f
3f
r,h
r,m
(r) −
(k)j3 (kr),(5.6)
ξ01,linµ − µ3
kdkP00,lin
= 1+
5b
5
2π 2

0
4

-1

real space

redshift space

2

1

s,m

-4

-6

1.4

1.2
s,m

ξ01 (r)/ξ01,lin(r)

-2
1.4

1

s,m

1.2

0

1

0.8
1

10
r [h-1Mpc]

100

10

20
r [h-1Mpc]

30

100

1

10

100

r [h-1Mpc]

r,mFigure 12. Upper panel of left set: mean pairwise momentum of dark matter in real space −2ξ01(r).
The dotted lines with the same color are the corresponding linear theory predictions, while the solidr,mlines show the Fourier transform of P01
measured from simulations. Lower panel of left set: theratios of the mean pairwise momentum to linear theory. For clarity, the results are offset by +30%,+20%, +10% and 0% from z = 2 to z = 0. Middle set: same as the left set but results for the dipoless,min redshift space −2ξ01(r). The solid lines are our model predictions based on nonlinear PT. To
clearly show both the upturn at small scales and large-scale asymptotic behavior, mixed linear andlogarithmic scales are used for the horizontal axis. Right panel : ratio of mean pairwise momentum inredshift space to that in real space. The dotted lines show linear theory predictions 1 + 3f /5 while
the solid lines are nonlinear PT results.

5.1.1

Dark matterIn the upper left panel of figure 12, we show the angularly-averaged, dipole component ofr,mmean pairwise momentum for dark matter in real space −2ξ01
measured using the estimator(equation (5.9)) at z = 0, 0.5, 1 and 2. Note that the velocity field is normalized by H as wasstated in section 3, so the mean pairwise momentum thus has a unit of h−1 Mpc. Becauser,mtwo objects moving towards each other has negative contribution, the sign of −2ξ01(r) isnegative at all scales. At all the four redshifts linear theory predictions, shown as the dottedlines, explain the trend of the N -body results up to ∼ 15 h−1 Mpc. In the lower left panelof figure 12, these results divided by the corresponding linear theory are plotted. For clarity
the results are offset by +30%, +20%, and +10% for results at z = 2, 1 and 0.5, respectively.
A careful look at the results reveals that the linear theory prediction for the measured mean
infall momentum starts to deviate from the N -body results at larger scales for lower redshifts,
as previously found [e.g., 18, 27]. In the both panels of the left set, the solid lines show ther,mFourier transform of −2P01
measured from simulations.
At the upper panel of the middle set of figure 12 we show the dipole moment of thes,mredshift-space mean pairwise momentum. The change of sign for −2ξ01(r) occurred at– 21 –

JCAP05(2014)003

1.6

s,m

r,m

z=2
1
0.5
0

-8

0

model
ξ01 (r)/ξ01 (r)sim linear

r,m
ξr,m
01 (r)/ξ01,lin(r)-2ξ01 (r) [h Mpc]

-1-2ξr,m01 (r) [h Mpc]

-2

This is the same estimator as that of equation (5.9), but the sum at the right-hand side istaken over each µ bin.
s,mWe plot the dark matter pairwise velocity in redshift space, −2ξ01(r, µ)/µ, at the topr,m(r), which is the same quantity as inrow in figure 13. For comparison we also plot −2ξ01s,m/µ is larger for separation perpendicular to the line-of-sight
figure 12. The amplitude of ξ01
s,m(r, µ = 0) and(µ = 0) than that along the line-of-sight (µ = 1). The difference between ξ01s,mthe real-space one is also larger than that between ξ01 (r, µ = 1) and the real-space one,s,m
opposite of our expectation in Fourier space. The N -body results of ξ01
for the separation
along the line-of-sight at the upturn scales are better predicted by our nonlinear PT thanthose perpendicular to the line-of-sight. In the second row of figure 13, the ratio of the 2D
pairwise streaming velocity in redshift space to the corresponding linear theory prediction,
s,m
s,m
s,m
ξ01
(r, µ)/ξ01,lin(r, µ) is shown. The deviation from linear theory is larger for ξ01
at µ = 0
than for µ = 1. At z = 0 the result for µ = 0.9 is consistent with linear theory at r ≥80 h−1 Mpc while that for µ = 0.1 agrees with linear theory at r > 100 h−1 Mpc. While
our PT prediction is in good agreement with the N -body measurements for larger µ, theprediction for the transverse separation is less accurately agreed with the measurement. In thebottom row of figure 13, we show the anisotropic pairwise streaming velocity in redshift spaces,mr,mdivided by the isotropic dipole in real space, ξ01(r, µ)/µ/ξ01(r). Because the anisotropy
is characterized by f (z), the higher-redshift results are more anisotropic. Again one can see
the significant deviation from linear theory for the mean pairwise momentum at µ = 0 evenat very large scales.
5.1.2

Halos

Now let us extend the analysis to biased objects, dark matter halos. The real-space meanr that was studied and the N -bodypairwise momentum is just a Fourier transform of P01– 22 –

JCAP05(2014)003

r ∼ 10 h−1 Mpc was seen in the recent observation [19]. This upturn can be seen in linear
theory, but the scale of the turn comes to much smaller scales and the change is also small.
On the other hand, the upturn is precisely predicted by nonlinear PT shown as the solidline. Note that while the amplitude of the mean infall momentum monotonically increases asredshift decreases in real space, it is not the case at low redshift in redshift space. It is because
the contribution of RSD to the mean streaming velocity is proportional to f 2 , not f . The
ratio of the measured mean infall momentum in redshift space to the corresponding linear
theory is presented at the bottom panel of the middle set of figure 12. The measurement
is more suppressed from the linear theory in redshift space than that in real space, because
the mean infall momentum in redshift space is affected by nonlinear velocity dispersion. It isinteresting to note that the deviation of the measurement in redshift space from linear theory
is larger even at large scales, around BAO scales of ' 105 h−1 Mpc. These trends in redshift
space are very well captured by the nonlinear PT. At the right panel in figure 12, the ratios
s,m
r,m
ξ01
(r)/ξ01(r) are shown. The linear theory predictions, that are (1 + 3f /5), are shown as
the horizontal lines with the same color. As expected the nonlinear PT has good agreementwith the N -body results.
Unlike the infall velocity in real space, the velocity in redshift space is expressed as theinfinite sum of odd powers of µ due to RSD, starting from µ1 , as we have seen in equation (5.5)
and also in equation (5.6) for the linear theory limit. In order to see the anisotropy on themean pairwise momentum due to RSD, we consider an estimator,Psss (r, µ)−2ξ01
i,j (vi · x̂i − vj · x̂j )cij
P 2.
(5.10)=µ
i,j cij,rand

0-1-2ξs,m01 (r,µ)/µ [h Mpc]

z=0

z=0.5

µ=0.1
0.5
0.9

-0.5

sim

µ=0.1
0.5
0.9

z=1
linear

z=2

µ=0.1
0.5
0.9

model

real space

-1

s,m
ξs,m
01 (r,µ)/ξ01,lin(r,µ)

1.2
1
0.8
0.6

r,m
ξs,m
01 (r,µ)/µ/ξ01 (r)

2
1.5
1
0.5
0
10

20

30

r [h-1Mpc]

40

100

10

20

30

40

100

r [h-1Mpc]

10

20

30

r [h-1Mpc]

40

100

10

20

30

40

100

r [h-1Mpc]

Figure 13. Top row : mean pairwise infall momenta of dark matter measured in redshift space,
s,m
−2ξ01
(r, µ)/µ for the three directions, µ = 0.1 (green), µ = 0.5 (red) and µ = 0.9 (blue). The realr,mspace dipole of the mean infall, −2ξ01(r), is also plotted as the black dashed line for comparison.
The linear theory and nonlinear PT predictions are shown as the dotted and sold lines with the samecolor, respectively. Middle row : the ratio of the redshift-space pairwise velocity to linear theory. For
clarity, the results for µ = 0.5 and µ = 0.9 are offset by +15% and +30%, respectively. Bottom row :the ratio of the mean streaming velocities in redshift space and in real space. Note that mixed linear
and logarithmic scales are used for the horizontal axis.

results were compared to perturbation theory in [38]. However, it is useful to show theconfiguration-space results because several previous works adopted the configuration-spacestatistics and also they can also be compared to the corresponding redshift-space statistics
we will present below. We thus show the real-space results in appendix A.
We present the pairwise mean infall momenta for halos in redshift space. At the top row
of figure 14 we show the angularly-averaged, dipole components. The results for dark matter,
presented already, are also shown for comparison. As is the case with dark matter, nonlinears,hPT at one-loop level can predict well the upturn in −2ξ01
seen at around r ∼ 10 h−1 Mpc.
s,hAt the second row we show the ratios of ξ01 from N -body simulations and nonlinear PT to
the corresponding liner theory. As we have already seen, redshift-space velocity statistics in– 23 –

JCAP05(2014)003

1.4

-1
-2ξs,h01 (r) [h Mpc]

0-2

redshift space

-4

z=0

-6

-8

linear

model

bin1
bin2
bin3
bin4
DM

sim

linear

z=1

model

bin1
bin2
bin3
bin4
DM

sim

linear

model

bin1
bin2
bin3
DM

s,h
ξs,h
01 (r)/ξ01,lin(r)

1.6
1.4
1.2
1
0.8
0.6
1.2

r,h
ξs,h
01 (r)/ξ01 (r)

1

0.8

0.6

0.4
50

100
-1

r [h Mpc]

50

100
-1

r [h Mpc]

50

100
-1

r [h Mpc]

s,hFigure 14. Top row : mean pairwise infall momenta of halos in redshift space −2ξ01(r). For
comparison the results for dark matter are shown as the black points. The dotted lines with thesame color are the corresponding linear theory predictions while the solid lines are the corresponding
nonlinear PT predictions. Second row : the ratios of the mean pairwise momenta to the corresponding
linear theory. The results are offset by +0%, +15%, +30% and +45% for halos from the lightest to
heaviest mass bins. Bottom row : the ratios of the mean pairwise momenta in redshift space and reals,hhspace, vmean(r)/vmean(r). The linear theory denoted by the dotted horizontal line gives the Kaiser
factor (1 + 3f /5b).

s and P s have much more nonlinear contributions than real-space statistics
Fourier space, P01
11
P01 and P11 . In configuration space there are significant nonlinear corrections up to verylarge scales through nonlinear transform from Fourier space [25]. Thus the deviation of thes,hredshift-space pairwise momentum ξ01
is much more significant at all the scales probed thanr,hthe real-space one ξ01. At scales r ≤ 100 h−1 Mpc the pairwise momentum for halos starts tobe suppressed relative to linear theory due to nonlinear velocity dispersion as is the case fordark matter. This large-scale deviation can be well explained by our nonlinear PT prediction.
For massive halo subsamples the amplitude of the measured momentum exceeds linear theory– 24 –

JCAP05(2014)003

-10
1.8

sim

z=0.5

r,h
ξs,h
01 (r,µ)/µ/ξ01 (r)

1.5

z=0
bin1

z=0.5
bin1

1

µ=0.1
0.5
0.9

z=1
bin1

sim

µ=0.1
0.5
0.9

model

µ=0.1
0.5
0.9

linearr,hξs,h01 (r,µ)/µ/ξ01 (r)

1.5
bin2

bin2

bin2

bin3

bin3

bin3

1

r,h
ξs,h
01 (r,µ)/µ/ξ01 (r)

1

10

100

10

r [h-1Mpc]

100r [h-1Mpc]

10100r [h-1Mpc]

Figure 15. The ratios of the mean pairwise momenta of halos in redshift space and the momentums,hr,hdipole in real space, ξ01(r, µ)/µ/ξ01(r). Results for z = 0, 0.5 and 1 are shown at the left, middle
and right columns, respectively. Linear and nonlinear predictions are respectively shown as the dottedand solid lines with the same color as the measurements.

at scales of r ∼ 30 h−1 Mpc and at smaller scales the amplitude goes below linear theorys,his also precisely predicted by nonlinear PT. We plot
again. This complicated behavior of ξ01
the ratio of the mean pairwise momenta between redshift space and real space at the bottom
row in figure 14. As in the right panel of figure 12, the horizontal lines are the linear theory
predictions, (1 + f (z)/3b1 ), where b1 is the large-scale constant halo bias (see section 3).
In order to see the angular dependence of the mean pairwise infall momentum due
to RSD, we show in figures 15 the ratio of redshift-space momentum to real space one asfunctions of (r, µ) measured using the estimator of equation (5.10). They are compared tothe prediction from linear and nonlinear PT, and good agreement of the N -body results tothe latter is found. As we have seen for dark matter in figure 13, linear theory has a poor
accuracy also for halos pairwise velocity in redshift space.
5.2

Momentum correlation function

The second fundamental velocity statistics in configuration space is an auto-correlation function of density-weighted peculiar velocity [5], the Fourier transform of the momentum auto
X . The momentum correlation is fully described in real space by the 3 × 3
power spectrum P11
correlation tensor Ψrij , and for a statistically homogeneous and isotropic velocity field, it
follows that [5, 61, 62]Ψrij (r) = h[1 + δ(xa )] [1 + δ(xb )] vi (xb )vj (xa )i = Ψ⊥ (r)δij + [Ψ∂ (r) − Ψ⊥ (r)] r̂i r̂j ,(5.11)

where Ψ∂ and Ψ⊥ are respectively the correlation functions parallel and perpendicular to theline of separation.2 The statistics for the momentum correlation can be defined using the2
Note that the subscript k is usually used in place of ∂, as Ψk , but in this paper k is used to describe aquantity along the line of sight.

– 25 –

JCAP05(2014)003

1.5

observable quantities, as [63, 64]3Xrẑi Ψrij (r)ẑj = pk (xb )pk (xa ) = ξ11(r).

(5.12)

i,jThus, the angular dependence of the momentum correlation function in real space is described as
2
2
r
r
r
(5.13)
ξ11(r) = ξ11,µ0 (r) + ξ11,µ2 (r)µ = Ψ⊥ (r) + [Ψ∂ (r) − Ψ⊥ (r)] µ .

(5.14)In redshift space we can use exactly the same definitions as in real space (equations (5.11)
and 5.12), so we can write,
D
E D
E
s
[1 + δ s (xa )] [1 + δ s (xb )] vks (xb )vks (xa ) = psk (xb )psk (xa ) = ξ11(r).
(5.15)
However, because of equation (5.1) in redshift space, the redshift-space velocity correlation
function is expressed as an infinite sum of the velocity-moment correlation functions,
X
s
s
j
(r) =
ξ11,µ
(5.16)
ξ11
j (r)µ .
j=0,2,···

As we have seen in the previous section, the velocity correlation in redshift space is equivalentrs(r).
(r) = ξ11,lin
to that in real space in the Kaiser limit, ξ11,lin
From N -body simulations, we measure the velocity correlation function, we use an
estimator introduced in real space by [63] and modified as,Pi,j (vi · x̂i )dij (vj · x̂j )XP
ξ11 (r) =,where
dij = x̂i · x̂j ,(5.17)
i,j dij,rand

−1 P
P
2
X = r or s and i,j d2ij,rand = 1 + ξ X (r)i,j dij . Under the distant observer approximation, simply dij = 1. Equation (5.17) gives the angularly-averaged, monopole moment of
the velocity correlation function. The estimator originally proposed in [63] is pair-weighted,
r (r), and will be discussed in appendix B.
related to our estimator as ψ r (r) = [1 + ξ r (r)]−1 ξ11
5.2.1

Dark matterIn the upper left panel of figure 16, we show the monopole momentum correlation function ofr,mdark matter in real space ξ11(r) for z = 0, 0.5, 1 and 2. The velocity correlation function has
a unit of [ h−1 Mpc]2 . Just like the redshift-space mean pairwise momentum, the momentum
correlation of dark matter does not simply evolve with redshift because the amplitude depends
on f 2 D2 . N -body results start to deviate from linear theory, denoted as the dotted lines, atthe scales r ∼ 10 h−1 Mpc for all the four redshifts. At smaller scales measurements fromsimulations exceed linear theory due to the nonlinear effects. The solid lines are the Fourierr,hr,mtransform of our prediction for P11: the scalar part of the dark matter P11
modeled using
N -body simulations and the vector part using nonlinear PT. Because the vector part waswell modeled using nonlinear PT, the prediction matches the N -body result in configuration– 26 –

JCAP05(2014)003

In linear theory, we have well-known results in real space [60, 62, 65];
ZZ
f2
j1 (kr)f2
r,mr,mΨ⊥ (r) = 2 dkP00 (k),Ψ∂ (r) = 2 dkP00(k)j0 (kr).
2π
kr
2π

sim linear

sim linear

model

10
0.9

real space

redshift space
1

2

2

0.8

r,mm
ξr,mm
11 (r)/ξ11,lin(r)

s,mm
ξs,mm
11 (r)/ξ11,lin(r)

1

1.5

1.5

1

1
0.7
1

10
-1

r [h Mpc]

100

1

10
-1

r [h Mpc]

100

10

100
r [h-1Mpc]

Figure 16. Upper panel of left set: angularly-averaged momentum correlation functions of darkr,mm
matter in real space ξ11(r) = hpm
k (x)pk (x + r)i. The dotted lines with the same color are thecorresponding linear theory predictions, and the solid lines are our nonlinear PT predictions. Lower
panel of left set: the ratios of the momentum correlation function to the linear theory. For clarity,
the results are offset by +75%, +50%, +25% and 0% for dark matter from z = 2 to z = 0. Middles,mset: same as left set but for redshift space, ξ11(r). Right panel : monopole-to-real-space ratio of thes,mr,mmomentum correlation functions, ξ11 (r)/ξ11 (r). The dotted line is the linear theory predictions,which is just unity while the solid lines with the same color as the points are corresponding nonlinear
PT predictions.

space up to small scales, r ∼ 5 h−1 Mpc at z = 0 and r ∼ 3 h−1 Mpc at z = 2. In the lower
left panel of figure 16, the momentum correlation function divided by the corresponding
linear theory is shown. At large scales the simulation results are systematically smaller thanlinear theory, almost by a constant [e.g., 18, 60]. Thus the larger scales we focus on and thesmaller the amplitude of the correlation becomes, the lager the deviation of the ratio to linear
theory deviates from unity becomes. On such large scales there is a small but non-negligible
improvement (∼ a few percent) by applying nonlinear PT. Recently a new method to model
the velocity correlation function based on a peculiar velocity decomposition was proposed
and tested by [66, 67].
At the upper panel of the middle set in 16, we show the momentum correlation function
for dark matter in redshift space, averaged over angle. Linear theory prediction in redshift
space shown as the dotted lines is exactly the same as that in real space, while it is not the casefor nonlinear PT shown as the solid lines. Although the cross power spectra between density
r do not contribute to ξ s , there are still many terms contributing
and velocity moments ξ0L
11rrr
to it such as ξ12 , ξ13 , ξ22 , and so on. Thus the measurements in redshift space start to exceed
linear theory at larger scales, particularly for lower redshifts. Nonlinear PT can explain themeasurements up to smaller scales than linear PT does, but it also breaks down at larger– 27 –

JCAP05(2014)003

r,mm
ξs,mm
11 (r)/ξ11 (r)

10

1

z=2
1
0.5
0

100

-2
2
ξs,mm
11 (r) [h Mpc ]

-2
2
ξr,mm
11 (r) [h Mpc ]

model

z=2
1
0.5
0

100

5.2.2

HalosFinally, let us extend the dark matter analysis to halos. We discuss the momentum correlar,hwith different mass bins at appendix A. At the top row of figure 18
tion function of halos ξ11
we show the angularly-averaged momentum correlation functions for halos in redshift space.
Unlike the case for the mean infall momentum, the shape of the momentum correlation function in redshift space is very similar to that in real space for all scales proved. It implies thateven the redshift-space momentum correlation at nonlinear scales is dominated by the realspace one that only contains linear-order contributions in the expansion (equation (2.20)).
Also the accuracy of nonlinear PT at small scales in redshift space is almost the same as forthat in real space. The bottom row of figure 18 compares N -body results to linear theory
and our nonlinear PT. Contrary to the real-space case, N -body measurements do not agreewith linear theory even up to very large scales, ∼ 70 h−1 Mpc for the halos with the smallest
mass and ∼ 100 h−1 Mpc at z = 0. On the other hand, nonlinear PT agrees with the N body results quite well. It is thus necessary to take into account the nonlinear corrections totheoretical models in order to predict the momentum correlation function in redshift space.
Figure 19 shows the ratio of the redshift-space and real-space momentum correlations,hr,hfunctions in 2D, ξ11(r, µ)/µ/ξ11(r). The ratio measured from simulations is compared tolinear theory that is unity and to nonlinear PT. The deviation of the ratio from unity due– 28 –

JCAP05(2014)003

scales than the case in real space. The ratio of the measured momentum correlation function
in redshift space to the corresponding linear theory is presented at the bottom panel of the
middle set of figure 16. The nonlinear, higher-order terms suppress the amplitude to deviate
from linear theory even at r < 90 h−1 Mpc for z = 0 and r < 50 h−1 Mpc for z = 2. One loop
PT dramatically improves the agreement with the N -body results, up to r < 20 h−1 Mpc fors,mr,mz = 0 and r < 4 h−1 Mpc for z = 2. At the right panel in figure 16, the ratios ξ11(r)/ξ11(r)
are shown. As is the case in Fourier-space analysis, in linear theory the ratio is just unity,
thus the deviations from unity are entirely from a nonlinear RSD effect. The effect alters theamplitude by at most ∼ 25% for z = 0 and ∼ 7% for z = 2. Due to the systematic deviationss,mr,mof ξ11and ξ11from the linear theory at large scales, there are slight discrepancies in theratios between N -body results and nonlinear PT at all scales. However, overall behaviors of
the nonlinear RSD are well explained using our one-loop PT.
r,m
We show the real-space momentum correlation function of dark matter in 2D, ξ11(r, µ),
at the first row of figure 17 as the dashed lines. We do not show the real-space result at z = 2
because it is very close to that in redshift space, as shown at the bottom right panel. The
results are well explained by linear theory, up to ∼ 20 h−1 Mpc at z = 0 and ∼ 10 h−1 Mpc
at z = 2, although there are small but systematic deviations for µ = 0.9 at larger scales atlower redshifts. The first row of figure 17 also shows the redshift-space momentum correlations,m(r, µ). Its linear theory prediction is equivalent to that in real space. Since infunction ξ11practice there are contributions from higher powers of µ due to RSD, the validity of linear
theory is limited to very large scales, while our nonlinear PT at one loop level captures thedeviation of the measurements from linear theory at r ≤ 10 h−1 Mpc. However, as shownin the second row of figure 17, there is still small disagreement between N -body results forµ = 0.9 and their nonlinear PT prediction. The bottom row of figure 17 plots the ratio
between momentum correlation functions in redshift space and in real space. Just like thecase of Fourier space in figure 3, the deviation of the ratio from unity is caused by nonlinear
RSD. At r > 10 h−1 Mpc the amplitude is suppressed by ∼ 30% at z = 0 and ∼ 10% at
z = 2.

-22
ξs,m
11 (r,µ) [h Mpc ]

z=0

z=0.5

z=1

z=2

10

1

sim

µ=0.1
0.5
0.9

linear

µ=0.1
0.5
0.9

model

real space
µ=0.1
0.5
0.9

s,m
ξs,m
11 (r,µ)/ξ11,lin(r,µ)

2

1.6

1.2

r,m
ξs,m
11 (r,µ)/ξ11 (r,µ)

0.8

1.2

0.8

10
r [h-1Mpc]

100

10

100

r [h-1Mpc]

10r [h-1Mpc]

100

10

100

r [h-1Mpc]

s,mFigure 17. Momentum correlation functions of dark matter in redshift space, ξ11(r, µ) (first row)for the three angular directions, µ = 0.1, 0.5 and 0.9 denoted by the green, red and blue points,r,mrespectively. The real-space result ξ11(r, µ) is shown as the dashed lines for comparison. The
second row shows the ratios of the redshift-space momentum correlation function to the linear theory,
s,m
s,m
ξ11
(r, µ)/ξ11,lin(r, µ). The results for µ = 0.5 and 0.9 are offset by +15% and +30%, respectively
for clarity. The bottom row shows the ratio of the momentum correlation function in redshift spaces,mr,mto that in real space ξ11(r, µ)/ξ11(r, µ). In all the panels the dotted and solid lines with the samecolor as the N -body measurements are the corresponding linear theory and nonlinear PT predictions,
respectively.

to nonlinear RSD is smaller for halos than for dark matter we have seen in figure 17, andbecomes smaller for more massive halos. Nonlinear PT agrees with the N -body results up
to r ∼ 10 h−1 Mpc for µ = 0.1 and 0.5 and up to r ∼ 30 h−1 Mpc for µ = 0.9.

6Effects of satellite galaxies on velocity statistics

So far we have focused on modeling velocity statistics for dark matter and halos using nonlinear PT. In real surveys we observe galaxies that have two types of contributions; central
galaxies which can be considered to move together with the host halos [but see, 68, 69], andsatellite galaxies that mainly exist in massive halos. Modeling the latter part is not a trivial– 29 –

JCAP05(2014)003

µ=0.1
0.5
0.9

bin1
bin2
bin3
bin4

-2
2
ξs,h
11 (r) [h Mpc ]

100

bin1
bin2
bin3
bin4

bin1
bin2
bin3

Redshift space
10

z=0.5

z=1

s,h
ξs,h
11 (r)/ξ11,lin(r)

1

2

1
1

10

100 1

-1

10

100 1

-1

r [h Mpc]

10

100

-1

r [h Mpc]

r [h Mpc]r,hξs,h11 (r,µ)/ξ11 (r,µ)

r,h
ξs,h
11 (r,µ)/ξ11 (r,µ)

r,h
ξs,h
11 (r,µ)/ξ11 (r,µ)

s,hFigure 18. Same as 27 but here we show the halo results in redshift space, ξ11
(r).

1.2

z=0

bin1

z=0.5

bin1

z=1

bin1

1
µ=0.1
0.5
0.9

0.8
1.2

bin2

bin2

bin2

bin3

bin3

bin3

1
0.8
0.6
2
1.5
1
0.5
0
10
r [h-1Mpc]

10010r [h-1Mpc]

100

10

100

r [h-1Mpc]

Figure 19. The ratios of the momentum correlation functions of halos in redshift space and ins,hr,h(r, µ)/ξ11(r, µ). Linear and nonlinear predictions are respectively shown as thereal space in 2D, ξ11dotted (unity) and solid lines with the same color as the measurements.

– 30 –

JCAP05(2014)003

z=0

6.1

Fourier space

We start the comparison in Fourier space. At the upper panel of the left set in figure 20
h of galaxies and halos which
we show the density-momentum power spectra in real space P01
have the same bias as the galaxies. The lower panel shows this cross power spectrum divided
r,h
r,h
r,h
measured for galaxies in real space is. The behavior of P01/P01,linby the linear theory, P01r,hconsistent to that for halos to k ≤ 0.14 h Mpc−1 for higher µ values. The amplitude of P01
forgalaxies becomes higher than that for halos at smaller scales, by ∼ 10% at k ≤ 0.2 h Mpc−1 .
This behavior has been already seen in [36].
The upper panel of the middle set in figure 20 shows the results for halos and galaxies
in redshift space and the lower panel shows their ratios to the corresponding linear theory.
The right panel in figure 20 shows the ratio of the redshift-space and real-space cross power
spectra. Because of the contributions from satellite galaxies, the galaxy sample has muchlarger velocity dispersion than halos. Thus the redshift-space density-momentum power
spectrum of galaxies is suppressed and changes the sign as is the case for the quadrupole
spectrum of galaxy density but at larger scales k ∼ 0.15 h Mpc−1 (see figure 5 of [36]).
Obviously, nonlinear PT for halos cannot predict the measured cross spectrum in redshift
space except for very large scales because we used linear theory to predict the velocitys,hfor galaxies is suppressed
dispersion of halos. At k ∼ 0.1 h Mpc−1 the amplitude of P01
relative to nonlinear PT for halos by ∼ 50%.
Figure 21 compares the results for the auto power spectra of momentum for halos andr,hgalaxies. The left panel presents the momentum auto spectrum in real space P11. As wediscussed in section 4.2.2 we assume the Poisson model for the shot noise for the momentum
power spectrum as hvk2 i/n. Thus there are systematic deviations between the measurements
and nonlinear PT, coming from the deviation from the Poisson model as well as the accuracyr,hof modeling the vector part of P11. In redshift space, as shown at the middle panel, theres,his a suppression of the auto power spectrum for galaxies P11(k, µ) for µ = 0.9 that cannot
be explained by the incorrect assumption of the Poisson model and thus can be considered
as nonlinear contributions from satellite galaxies. However, like the case for the cross powers,hshown above, P11
for galaxies is relatively well modeled by nonlinear PT at one-loop level
predicted for halos, compared to linear theory. The ratio of the redshift-space and reals,hr,hspace power spectra of momentum, P11(k, µ)/P11(k, µ), is shown for halos and galaxies at– 31 –

JCAP05(2014)003

task, which makes it hard to model the statistics of a galaxy sample. It is thus meaningful
to test whether the velocity statistics we considered above can be modeled for galaxies using
PT of halos.
As a mock galaxy catalog, we use a halo occupation distribution (HOD) modelingwhich populates dark matter halos with galaxies according to the halo mass [e.g., 70–73].
We consider as a mock galaxy sample the galaxies from the Baryon Oscillation Spectroscopic
Survey (BOSS) [74, 75], referred to as the “CMASS” galaxy sample. Galaxies are assigned
to the halos at z = 0.5 using the best fit HOD parameters for CMASS galaxies determinedby [76]. For halos which contain satellite galaxies, we randomly pick up the same number ofdark matter particles to represent the positions and velocities of the satellites. This method
was applied in our previous work [36] and the good agreement with the observation has
been confirmed. In the following we will compare the real-space and redshift-space velocity
statistics measured from the mock galaxy sample to the nonlinear PT prediction for the halosubsample that has the same bias as the galaxy sample (bin2 at z = 0.5).

102
µ=0.1
0.3
0.5
0.7
0.9

Real space
s,hh
Ps,hh
01 /P01,no-wiggle

101
1

1.5

1

1

0.5

0.5
0

0.05

0.1

0.15

0.2

0.25

0.3

0.05

0.1

-1

0.15

0.2

0.25

0.3

0.05

0.1

-1

k [h Mpc ]

0.15

0.2

0.25

0.3

k [h Mpc-1]

k [h Mpc ]Figure 20. At the upper panel of the left set we show the density-momentum power spectrum in realr,hspace k 2 P01
at z = 0.5, and at the lower panel we plot its ratio to the corresponding linear theoryr,hr,hprediction without BAO, P01/P01,no-wiggle. The middle set is the same as the left set but for the
results in redshift space. The right panel shows the ratio of the density-momentum spectra in redshifts,hr,hspace and in real space, P01/P01. The colored points show the results for halos at z = 0.5 and theblack points with the same point types as the colored points show the results for mock galaxies withthe same values of µ. The halo subsample has the same bias as the galaxy sample. The dotted lines
show the linear Kaiser model while the solid (positive) and dashed (negative) lines show our one-loop
PT prediction. For clarity we show only results for some µ bins among the five µ bins.

2000

s,hh

µ=0.1
0.5
0.9
0.05

0.1

0.15

0.2

0.25

k [h Mpc-1]

0.3

1

r,hh

2000

P11 (k,µ)/P11 (k,µ)

k2Ps11(k,µ)

Redshift space4000

k2Pr11(k,µ)

Real space
4000

0.05

0.1

0.15

0.2

k [h Mpc-1]

0.25

0.3

0.8

0.05

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 21. Same as figure 20 but results for the momentum auto power spectra are shown. The power
spectra in real and redshift space, respectively shown in the left and middle panels, are multiplied by
k 2 for clarity.

the right panel of figure 21. As expected, the deviation of the ratio from unity caused by
nonlinear RSD is larger for galaxies that have contributions from satellites than for halos.
6.2

Configuration space

Finally let us consider effects of satellite galaxies on velocity statistics in configuration space,
the mean pairwise infall momentum and the momentum correlation function. The upper
panel of the left set in figure 22 shows the mean pairwise momentum of galaxies measured in– 32 –

JCAP05(2014)003

r,hh
Pr,hh
01 /P01,no-wiggle

101

r,hh
Ps,hh
01 (k,µ)/P01 (k,µ)

k2Ps01(k,µ)

102

2 r

k P01(k,µ)

Redshift space

1

4
2

redshift space
0.5

0
-2
-4

Ratio to linear

Ratio to linear

mean infall momentum
1
1.8
1.6
1.4
1.2
1

6

r,h
ξs,h
01 (r)/ξ01 (r)

-1
2ξr,h
01 (r) [h Mpc]

10-1-2ξs,h01 (r) [h Mpc]

8galaxies
halos (bin2)
linearmodel
real space

100

1
0.8
0.6

0

-0.5

0.4

10
r [h-1Mpc]

10

20

30
r [h-1Mpc]

40

50

60

1

10
r [h-1Mpc]

Figure 22. From the left, we show the mean pairwise infall momenta in real space, those in redshift
space, and the ratios of the mean infall in redshift space and in real space. At the left panel, we show
r,h
r,h
r,h
has a negative sign for all the scales. The black and red points
because −2ξ01) = 2ξ01−(−2ξ01
respectively show the results for mock galaxies and halos at z = 0.5. The halo subsample has almost
the same bias as the galaxy sample. The dotted lines show the linear Kaiser model while the solidlines show our PT prediction for the halo subsample.
r,hreal space at z = 0.5. Because this statistics −2ξ01
has a negative sign at all the scales, wer,h r,hr,hr,h/ξ01,lin . We also
plot −(−2ξ01 ) = 2ξ01 . The lower panel shows its ratio to linear theory ξ01
compare it to the measurement and one-loop nonlinear PT for halos. The measured velocity
of galaxies deviates from the prediction for halos at slightly larger scales, r ' 10 h−1 Mpc,
but for r > 10 h−1 Mpc the mean pairwise momentum for galaxies is almost equivalent tor,hthat for halos. While the amplitude of ξ01
for halos is suppressed at small scales because offinite sizes of halos, that is not found for the galaxy sample because of the one-halo term.
The middle set in figure 22 is the same as the left but shows the results in redshift spaces,h−2ξ01 . The result for mock galaxies starts to deviate from the nonlinear PT for halos at
one-loop level at r ' 40 h−1 Mpc. Because of the FoG effect from satellite galaxies, the scaleof upturn that appeared at r ' 10 h−1 Mpc for halos shifts to larger scales, r ∼ 20 h−1 Mpc,
and is no longer predictable with the halo nonlinear PT. Thus, in order to predict the upturn
scale of the mean infall velocity measured by [19] for detection of the kSZ effect, that is also
at r ∼ 20 h−1 Mpc, one needs to properly take into account the effect of the FoG, that shouldbe pursued in future work. The ratio of the mean pairwise momentum in redshift space tothat in real space is shown in the top right panel in figure 22.
Figure 23 present the comparison of the momentum correlation function for galaxies tothat for halos. The upper panel of the left set shows the real-space momentum correlationr,hfunctions for halos and galaxies, ξ11, and the lower panel shows their ratios to the linear
theory prediction. The real-space result for galaxies is similar to that for halos, down to lessthan 10 h−1 Mpc. The middle panels in figure 23 are the same as the left panels but shows,hthe results in redshift space ξ11. The result for galaxies slightly deviates from nonlinear PT
at one loop at r ≤ 30 h−1 Mpc, but the trend of the shape of the correlation function is still
well described with the nonlinear PT down to scales similar to the result in real space. The
ratio of the momentum correlation function in redshift space to that in real space is shownin the right panel of figure 23. While the effect of nonlinear RSD on the halo momentum
correlation is ∼ 10% that is predicted by our nonlinear PT at r ≥ 5 h−1 Mpc, that on thegalaxy momentum correlation has a ∼ 20% effect. Taking into account the contribution– 33 –

JCAP05(2014)003

0.2
1real space

101

102
1.2
redshift space
r,h
ξ s,h
11 (r)/ξ 11 (r)

-2
2
ξ r,h
11 (r) [h Mpc ]

102

-2
2
ξ s,h
11 (r) [h Mpc ]1.4
galaxies
halos (bin2)
linear
model

101
10

Ratio to linear

Ratio to linear

momentum correlation
10

5

5
0.8
0

1

10
-1

r [h Mpc]

1

10
-1

r [h Mpc]

1

10
-1

r [h Mpc]

Figure 23. Same as figure 22 but results for the momentum correlation functions are shown.

of satellite galaxies is also an important issue to model a redshift-space power spectrum ofgalaxies. The effect of satellite galaxies needs to be studied for a theoretical modeling of
velocity statistics in redshift space.

7

Conclusions

In this paper we have studied the cross power spectrum between the density and momentum
density fields together with the auto power spectrum of the momentum density field, which
are the fundamental statistics being measured in peculiar velocity surveys. What we candirectly measure from velocity surveys is the momentum density field in redshift space, rather
than the velocity field in real space, which was the focus of most of the previous analyzes. This
leads to corrections both at the linear and nonlinear level. To model them, we extended thephase-space distribution function approach originally developed to predict the redshift-space
power spectrum of the density field [34–38]. In doing so, we showed that the redshift space
momentum field can be expressed in terms of a sum over mass-weighted velocity moments inreal space, starting from the real space momentum density field. Likewise, the momentum
power spectra in redshift space can be expressed in terms of correlators between the Fouriercomponents of these moments, analogously to the density-density power spectrum.
We derived momentum power spectra using both the linear Kaiser model [3] and theperturbative approach developed in [37, 38]. To test the accuracy of the linear theory andphase space approach, we compared the theoretical predictions for the density-momentum
cross power and momentum auto power spectra in redshift space with those measured fromcosmological N -body simulations. We found that, for dark matter, the velocity dispersion
affects the redshift-space cross power spectrum between the density and momentum. Therefore, for dark matter the numerical results deviate from linear theory at k ∼ 0.1 h Mpc−1
by ∼ 50% at z = 0. Our nonlinear PT prediction, which relies on the real-space dark matter power spectrum measured from the simulations, achieves a much better agreement thanlinear PT. It can reproduce the N -body results at z = 0 for the transverse direction µ ∼ 0.3
nearly perfectly up to k ' 0.3 h Mpc−1 , and for the line-of-sight direction µ ∼ 0.9 up to
k ' 0.06 h Mpc−1 . The accuracy of our nonlinear PT is even better for the momentum
auto power spectrum, because it is less affected by the velocity dispersion. It predicts the– 34 –

JCAP05(2014)003

0

1

Acknowledgments
We would like to thank Tobias Baldauf and Beth Reid for useful discussions and Stephen
Appleby for careful reading of this manuscript. We also thank the anonymous referee for many
useful suggestions. This research was supported by the DOE, WCU grant R32-10130, Ewha
Womans University research fund 1-2008-2935-001-2, and the Swiss National Foundation
under contract 200021-116696/1. V.D. acknowledges support by the Swiss National Science
Foundation.

A

Real-space velocity statistics of halos

Peculiar velocity statistics were studied in real space in detail in previous works. The main
purpose of this paper is to present the formalism and numerical analysis for the peculiar velocity statistics in redshift space. However, it is still useful to present the real-space velocities
to compare to the redshift-space velocity. Thus in this appendix we present the real-spacevelocity statistics for halos.
A.1

Fourier space

Because we studied the density-momentum power spectrum of halos in real space in detail
r,h
r,h
r,h
in [36, 38], the ratio of P01
to the corresponding linear theory, P01/P01,no-wiggle
is shownr,honly for z = 0 in figure 24. As we have seen in [36], the excess of the amplitude of P01
due
to the nonlinear effects is larger for more massive halos. Since we use the best fit values of b1r,hand b2 found in [38], our prediction for P01
agree perfectly with the N -body results, underr,mr,mthe condition that the dark matter spectra P01
and the scalar part of P11
are perfectlymodeled. The ratios of the real-space momentum spectra to the corresponding linear nowiggle spectra are presented at the first, second and bottom rows in figures 25 for z = 0,r,h0.5, 1, respectively. At all the redshifts, P11
is well explained by our model for less massive– 35 –

JCAP05(2014)003

N -body measurement for dark matter at z = 0 for the line-of-sight direction µ ∼ 0.9 up to
k ' 0.08 h Mpc−1 .
Because the dark matter halos have smaller peculiar velocity than the dark matterparticles, the momentum power spectra for halos are better modeled than those for dark
matter by combining nonlinear PT at one-loop level with a physically motivated descriptionr,hs,hof halo biasing. However, the auto power spectra of the momentum field P11
and P11areaffected by exclusion and other nonperturbative effects that require furhter understanding inorder to precisely model the momentum auto power spectrum of sparse, biased objects.
We have also presented an analysis of peculiar velocity statistics in configuration space.
We measured the mean infall pairwise momentum and the momentum correlation function
for both dark matter and halos. We then compared these measurement in real and redshift
space to our one-loop PT calculations, finding significant improvements relative to linear
theory. As an example of a realistic galaxy sample, we constructed a mock BOSS-typegalaxy sample by applying the HOD modeling to the simulated halos at z = 0.5. Since theinclusion of the effect of satellite galaxies on velocity statistics is not straightforward within
our PT formalism, we simply tested how well our nonlinear PT of halos can predict peculiar
velocity statistics of galaxies. We found that a more detailed PT model is required to improve
the agreement.

r,h
Pr,h
01 (k,µ)/P01,no-wiggle(k,µ)

4Real space
bin1

µ=0.3
0.5
0.9

2

sim
model
linear

bin2

bin3

1

0.05

0.1

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3 0.05

0.1

k [h Mpc-1]

0.15

0.2

0.25

0.3

k [h Mpc-1]

r,h
Pr,h
11 (k,µ)/P11,no-wwigle(k,µ)
r,hPr,h11 (k,µ)/P11,no-wiggle(k,µ)
r,hPr,h11 (k,µ)/P11,no-wiggle(k,µ)

Real space
bin1

z=0
bin2

bin3

10

µ=0.1
0.3
0.5
0.9

1

sim

µ=0.1
0.3
0.5
0.9

model
linear

z=0.5
bin1

bin2

bin3

bin2

bin3

10

1
z=1
bin1
10

1
0.05

0.1

0.15

0.2

0.25

k [h Mpc-1]

0.3 0.05

0.1

0.15
k [h Mpc-1]

0.2

0.25

0.3 0.05

0.1

0.15

0.2

0.25

0.3

k [h Mpc-1]

Figure 25. Momentum auto power spectra for halos in real space divided by the no-wiggle spectrar,hr,h/P11,no-wiggle
at z = 0 (top row), z = 0.5 (second row) and z = 1 (bottom row).
P11

halos at lower redshifts. However, there are systematic deviations of N -body results for more
massive halos from the corresponding PT due to the assumption of the Poisson shot noise.
A.2

Configuration space

The real-space mean pairwise momenta of halos are shown at the top row in figure 26,
compared to those of dark matter denoted as the black points. Because the mean pairwise– 36 –

JCAP05(2014)003

Figure 24. Density-momentum cross power spectra for halos in real space divided by the no-wiggler,mr,mspectra at z = 0, P01/P01,no-wiggle. The corresponding linear theory predictions and our model, bothof which are isotropic, are shown as the dotted and solid lines, respectively.

100

-1
2ξr,h
01 (r) [h Mpc]

Real space

10
z=0

z=0.5

z=1

1
bin1
bin2
bin3
bin4
DM

bin1
bin2
bin3
DM

r,h
ξr,h
01 (r)/ξ01,lin(r)

0.1
1.8
1.6
1.4
1.2
1
0.8
1

10
-1

r [h Mpc]

100

1

10
-1

r [h Mpc]

100

1

10

100

-1

r [h Mpc]

r,hFigure 26. Top panels: mean pairwise infall momenta of halos in real space. Because −2ξ01
has a
r,h
r,h
negative sign at all the scales, we plot −(−2ξ01 (r)) = 2ξ01 (r). For comparison the results for dark
matter are shown as the black points. The dotted lines with the same color are the correspondingr,hlinear theory predictions while the solid lines are the Fourier transform of the prediction for 2P01.
Bottom panels: the ratios of the mean pairwise momentum to the corresponding linear theory. For
clarity, the results are offset by +0%, +15%, +30% and +45% for halos from the lightest to heaviest.

r,h
r,h
r,h
) = 2ξ01
has a negative sign for all the scales, we plot −(−2ξ01
momenta in real space −2ξ01with a logarithmic scale. More massive halos have larger amplitude for the pairwise streaming
momentum because it is proportional to bias. At the bottom row we show their ratios tor,hlinear theory predictions. At z = 0, 2ξ01
of smaller halos starts to deviate from linear theory
at larger scales, but that of larger halos diverges at larger scales, as we have seen in different
statistics [e.g., 27, 36, 77]. At z > 0 the deviation from linear theory starts at larger scales
for more massive halos (∼ 40 h−1 Mpc for bin4 halos at z = 0.5) than for less massive halos(∼ 20 h−1 Mpc for bin1 halos). The solid lines are the Fourier transform of the predictionr,hwe discussed in previous section. It agrees with the N -body measurements for allfor −2P01
the four redshifts and for the scales much smaller than r ∼ 10 h−1 Mpc.
In figure 27 we present the monopole of the momentum correlation functions of halosr,hξ11(r) with different mass bins at redshifts z = 0, 0.5, 1 and 2. Since the velocity correlation
does not depend on halo bias in linear theory, we multiply the amplitude by 1.4, 1.42 and 1.43
for bin 2, 3 and 4 results, respectively, for clarity at the top panels. N -body results for the
momentum correlation function are well explained within linear theory at large scales. The
result for more massive halos starts to deviate from linear theory at slightly larger scales,∼ 30 h−1 Mpc for the most massive halos, while ∼ 20 h−1 Mpc for low mass halos. Ourmodel predicts the simulation results down to much smaller scales. However, because of thefinite size of halos, the amplitude of the measured momentum correlation is suppressed atsmall scales. It starts at larger scales for more massive halos that have larger size. As is– 37 –

JCAP05(2014)003

bin1
bin2
bin3
bin4
DM

bin1
bin2
bin3
bin4

-2
2
ξr,h
11 (r) [h Mpc ]

100

bin1
bin2
bin3

bin1
bin2
bin3
bin4

Real space
10

z=0

z=1

z=0.5

r,h
ξr,h
11 (r)/ξ11,lin(r)

2

1
1

10

100 1

r [h-1Mpc]

10r [h-1Mpc]

100 1

10

100

r [h-1Mpc]

r,hFigure 27. (Top) Momentum correlation functions of halos in real space ξ11(r). The dotted and solidlines show the corresponding linear and nonlinear PT predictions, respectively. Since the correlation
does not depend on bias in linear theory, the amplitude is multiplied by 1.40 , 1.41 , 1.42 and 1.43 forbin 1, 2, 3 and 4 results for clarity. (Bottom) The ratios of the momentum correlation function to
the corresponding linear theory. For clarity, the results are offset by 0%, +25%, +50% and +75% forhalos from the lightest to heavies.

the case for dark matter presented in figure 16, the amplitude of the momentum correlation
is systematically smaller than linear theory at large scales as seen in the bottom panels of
figure 27. The trend can be seen in the work by [60] and [18] that was based on linear theory
and a halo model, respectively. Nonlinear PT improves the accuracy at the large scales,
particularly for massive halos by ∼ 5%, but the measured amplitude is still systematicallyhigher.

B

Density-weighted vs pair-weighted velocities

In this paper we focus on velocity statistics based on density-weighted velocity. It is worth
comparing the accuracy of the nonlinear PT for density-weighted velocity statistics to thatfor pair-weighted velocity statistics, which are often used in literature [e.g., 19, 20, 25, 56].
B.1

Mean pairwise velocityrs
The pair-weighted pairwise velocity in real space vkmean
and in redshift space vkmean
isgiven by
D
iE

h
1 + δ X (xa ) 1 + δ X (xb ) vkX (xb ) − vkX (xa )
X (r)
−2ξ01
X
vkmean
(r) ≡
=
,
(B.1)
h[1 + δ X (xa )] [1 + δ X (xb )]i
1 + ξ X (r)– 38 –

JCAP05(2014)003

1

0

-2

-4

z=0

-6

DM
bin1
bin2
bin3
bin4

2

Redshift space

z=0.5

z=1

DM
bin1
bin2
bin3
bin4

DM
bin1
bin2
bin3

0

-2

-4

20
r [h-1Mpc]

40

20
r [h-1Mpc]

40

20

40

r [h-1Mpc]r(r) =Figure 28. Pair-weighted mean pairwise velocities for dark matter and halos in real space vmean
r
r
−1
s
s
s
−1
ξ01 (r)[1+ξ (r)] (top) and in redshift space vmean (r) = ξ01 (r)[1+ξ (r)] . The dashed lines with thesame color are the corresponding linear theory predictions while the solid lines are the corresponding
nonlinear PT predictions.

where r = xb − xa . As is clear from equation (B.1), in order to predict the pair-weighted
pairwise velocity in real and redshift space one needs to additionally model the density
correlation function in real and redshift space, respectively, that comes into the denominator.
The effect of the denominator becomes more significant for a sample with the higher bias.
It is one of the disadvantages to use the pair-weighted velocity. Moreover, the pair-weighted
X
velocity is not equivalent to volume-weighted velocity, namely vkmean
6= −2hδ X (xb )vkX (xa )i.
r,hThe dipole component of the pair-weighted mean infall velocity in real space, vmean(r),measured using the estimator of equation (5.7), is shown in the top row of figure 28. For
linear theory denoted by the dashed lines, the correlation function in the denominator isr,hr,malso computed using linear theory ξlin
= b21 ξlin
. The agreement of the N -body result tolinear theory differs from the case for the density-weighted velocity by a factor of (1 +r,m
ξ r,h (r))/(1 + b21 ξlin(r)). At small scales where the N -body results largely deviate from linear−1
theory (r < 10 h Mpc), the deviation becomes smaller for the pair-weighted mean infall
r and that in ξ r cancel out to some extent.
velocity at a glance because the nonlinearity in ξ01
However, it does not mean that the pair-weighted velocity is better modeled by linear theory
than density-weighted velocity. The black solid lines are the predictions for dark mater,
r,m
r,m
vmean
= −2ξ01/[1 + ξ r,m ], where ξ r,m is also measured from simulations, thus the accuracy is
the same as the solid lines at the top panel of the left set in figure 12. The solid curves with– 39 –

JCAP05(2014)003-1
vs,hmean(r) [h Mpc]

-1vr,hmean(r) [h Mpc]

Real space

B.2

Velocity correlation function

The velocity correlation function, studied in the previous section, can also be defined similarlyas pair-weighted statistics. It is expressed in real space ψ r and in redshift space ψ s as
ψ X (r) =

2 X
X (r)
(1 − µ2 )ΨXξ11⊥ (r) + µ Ψ∂ (r)=,h[1 + δ X (xa )] [1 + δ X (xb )]i
1 + ξ X (r)(B.2)

X , modelingwhere X = r or s. Compared to the density-weighted velocity correlation ξ11
the pair-weighted correlation ψ X requires the additional modeling of the density correlation
function ξ X .
First we focus on the velocity correlation measured in real space ψ r . [65, 78] claimed
that linear theory fails to predict the real-space velocity correlation function even at verylarge scales, but later it was pointed out by [60] that the disagreement was because of theconfusion of the pair-weighted and density-weighted velocity statistics. As shown at the
top row of figure 29, the pair-weighted velocity correlation function of halos measured fromsimulations in real space ψ r agrees with linear theory up at r > 20 h−1 Mpc at z = 0 andr > 40 h−1 Mpc at z = 1, confirming the work by [60]. This accuracy is more or less thesame as the case for the density-weighted velocity correlation in real space shown in figure 27.
Our model predicts the N -body results in real space with similar accuracy to the case for thenumber-weighted velocity correlation because the real-space halo correlation function ξ r,h isr,mprecisely modeled with the perfect modeling of P00
and the nonlocal bias model.
Next we present the results of the pair-weighted velocity correlation function measured
in redshift space ψ s at the bottom row of figure 29. Note that the corresponding linearsrtheory in redshift space is no longer the same as that in real space, ψ11,lin
6= ψ11,lin
despitesr
s22r
ξ11,lin = ξ11,lin because ξlin = (b + f µ ) ξlin . Good agreement between numerical results andone-loop nonlinear PT is found for both dark matter and halos for the similar range to thecase for the number-weighted velocity correlation shown in figures 16 and 18. However, at thescales lower than the valid range of nonlinear PT, the disagreement in the halo correlation– 40 –

JCAP05(2014)003

r,hother colors are the predictions for halos, vmean, where the measured ξ r,m and the nonlocal
bias model are used [38]. In figure 26 we have seen that the measured density-weighted
pairwise velocity for halos in real space is accurately modeled down to r < 10 h−1 Mpc.
On the other hand, that of the pair-weighted velocity agrees with N -body results only atlarger scales. Particularly for the most massive halo subsamples, the disagreement starts atr ∼ 30 h−1 Mpc at z = 0 and 0.5.
The dipole component of the pair-weighted mean pairwise velocity in redshift space,ss (r)[1 + ξ s (r)]−1 , is the statistics used by [19] to claim the detection of thevmean(r) = −2ξ01skSZ effect. The bottom row of figure 28 shows the measurements of vmeanusing the estimatorof equation (5.7). Here we model the redshift-space correlation function ξ s using nonlinearr,m
PT in the same way as [38]. Briefly, in equation (2.8) we use the N -body results for P00
,
r,m
r,m
4
P01 and the scalar part of P11 , nonlinear PT for the other terms up to µ . Then fordark matter we use the best fit parameters for the nonlinear velocity dispersion as we dids,ms,mfor the redshift-space momentum power spectrum P01
and P11, while for halos we use thesimple model of the linear theory velocity dispersion and adopt the nonlocal bias model. The
N -body results for the pair-weighted mean pairwise velocity are predicted by the nonlinear
PT for both dark matter and halos, as well as those for the density-weighted velocity shownin figure 14, up to the peak scales at ∼ 20 h−1 Mpc.

DM
bin1
bin2
bin3
bin4

ψr,h(r) [h-2Mpc2]

10

DM
bin1
bin2
bin3
bin4

z=0

DM
bin1
bin2
bin3

z=0.5

z=1

1
Real spaceψs,h(r) [h-2Mpc2]

1
Redshift space

110
r [h-1Mpc]

100

1

10
r [h-1Mpc]

100

1

10

100

r [h-1Mpc]

Figure 29. Same as figure 28 but the pair-weighted velocity correlation functions in real spacersψ r (r) = ξ11(r)[1 + ξ(r)]−1 (top) and in redshift space ψ s (r) = ξ11(r)[1 + ξ s (r)]−1 (bottom). The−1
amplitude is multiplied by 1.4 for results for dark matter, and by 1.40 , 1.41 , 1.42 , and 1.43 for halos
with from the lightest mass (bin1) to the largest mass (bin4).

becomes large very quickly because of large nonlinearity of halo clustering ξ s,h . Also noteagain that the pair-weighted velocity correlation function defined by equation (B.2) is not
equivalent to the volume-weighted velocity correlation function ψ X (r) 6= hvkX (xb )vkX (xa )i.
We thus conclude that there is no advantage in defining pair-weighted velocity statistics thatare commonly used in previous works.

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