Mon. Not. R. Astron. Soc. 000, 1–16 (0000)

Printed 17 January 2011

(MN LATEX style file v2.2)

The pairwise velocity probability density function inmodels with local primordial non-Gaussianity
Tsz
Yan Lam1⋆ , Takahiro Nishimichi1⋆ & Naoki Yoshida1⋆
1

Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8583, Japan

17 January 2011

ABSTRACT

We study how primordial non-Gaussianity affects the pairwise velocity probability
density function (PDF) using an analytical model and cosmological N -body simulations. We adopt the local type non-Gaussian models characterized by fnl , and examine
both the linear velocity difference PDF and the linear pairwise velocity PDF. We show
explicitly how fnl induces correlations between originally independent velocities along
the parallel and the perpendicular to the line of separation directions. We compare
the model results with measurements from N -body simulations of the non-Gaussianmodels. Linear theory fails to predict the PDF in the fnl models. Therefore we develop
an analytic model based on the Zeldovich approximation to describe the evolution of
the velocity PDF. Our analytical model and simulation results show remarkably good
agreement in both the parallel and the perpendicular directions for the PDF profiles,
as well as the change in the PDF due to primordial non-Gaussianity. The agreement isparticularly good for relatively small separations (< 10h−1 Mpc). The inclusion of theevolution of the velocity PDF is important to obtain a good description on the signature of primordial non-Gaussianity in the PDF. Our model provides the foundation
to constrain fnl using the peculiar velocity in future surveys.
Key words: methods: analytical - dark matter - large scale structure of the universe

1

INTRODUCTION

Primordial non-Gaussianity has attracted much attention owing to its ability to distinguish inflationary models (e.g., Buchbinder et al. 2008; Khoury & Piazza 2008; Silvestri & Trodden 2008; Wands 2010, and references therein). Several cosmologicalprobes using the CMB (Komatsu et al. 2010; Hikage et al. 2008; Yadav & Wandelt 2008; McEwen et al. 2008; Rossi et al.
2009; Smidt et al. 2010; Komatsu 2010; Rossi et al. 2010) and the large-scale structures in the Universe (Koyama et al. 1999;
Matarrese et al. 2000; Scoccimarro et al. 2004; Sefusatti & Komatsu 2007; Izumi & Soda 2007; Lo Verde et al. 2008; Dalalet al. 2008; Matarrese & Verde 2008; Carbone et al. 2008; Afshordi & Tolley 2008; Slosar et al. 2008; McDonald 2008; Taruya
et al. 2008; Slosar 2009; Grossi et al. 2008; Kamionkowski et al. 2009; Desjacques et al. 2009; Pillepich et al. 2008; Lam &Sheth 2009; Grossi et al. 2009; Lam et al. 2009, 2010; Nishimichi et al. 2009; Cunha et al. 2010; Sartoris et al. 2010; Desjacques& Seljak 2010a; Liguori et al. 2010; Tseliakhovich et al. 2010; Xia et al. 2010; Chongchitnan & Silk 2010; Giannantonio &Porciani 2010; Marian et al. 2010; Smith et al. 2010) have been considered.
Here, we introduce a new large-scale structure probe of primordial non-Gaussianity – the pairwise velocity PDF. Current
large-scale structure probes of primordial non-Gaussianity focus on the clustering of halos, halo/void abundances, bispectrum,and the PDF of dark matter field – all of them are related to the change of the density field due to primordial non-Gaussianity.
It is important to note that both the initial density field and the initial velocity field are generated from the primordial
perturbation, and that their linear relation is described by the continuity equation. Hence, one may naively expect that
primordial non-Gaussianity affect the velocity field as well as the density field. While various measurements associated withthe change of density field have been extensively studied, there are few studies on the effect of primordial non-Gaussianity
on the velocity field. Scherrer (1992); Catelan & Scherrer (1995) discussed the effect of primordial non-Gaussianity on the⋆E-mail:tszyan.lam@ipmu.jp

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T. Y. Lam, T. Nishimichi & N. Yoshidadistributions of linear velocity fields. In particular Catelan & Scherrer (1995) studied the distribution of the parallel componentof the pairwise velocity using linear theory. Lam et al. (2010) discussed the effect of primordial non-Gaussianity on the redshift
space distortion. They made use of the ellipsoidal collapse model to derive the modification in the Kaiser factor relating thereal and redshift space power spectra, without discussing the modification in the velocity field. During the preparation of thiswork, Schmidt (2010) studied the primordial non-Gaussianity signature in the peculiar velocities of density peaks using linear
theory. Our present study has two important improvements over these previous works: first we show that primordial nonGaussianity induces a correlation betweeen velocities in the parallel and the perpendicular to the line of separation directions.
We show explicitly how this correlation, which is absent in Schmidt (2010), modifies significantly the linear velocity PDF.
Secondly, we show that the linear theory does not provide a good description of the signature of fnl in the pairwise velocity
PDF even at separation as big as 50 h−1 Mpc. We use cosmological N -body simulations to show this. We thus develop ananalytic model to describe the evolution of the velocity PDF. Our model is based on the Zeldovich Approximation. We
illustrate the improvement in the N -body measurement comparisons.
The peculiar velocity field has been investigated as a probe of cosmology for both dark matter (Gorski 1988; Seto &Yokoyama 1998; Sheth et al. 2001; Kuwabara et al. 2002; Scoccimarro 2004) and biased tracers (Sheth & Diaferio 2001;
Sheth et al. 2001; Hamana et al. 2003; Sheth & Zehavi 2009). Other studies (Yoshida et al. 2001; Peel 2006; Bhattacharya& Kosowsky 2007, 2008) discuss the possibility of constraining dark energy via the kinetic Sunyaev-Zeldovich effect. In thisstudy we focus on the peculiar velocity field of the dark matter field. The effect of primordial non-Gaussianity on the velocity
of biased tracers will be discussed in future work.
We first describe how the linear velocity PDF changes due to primordial non-Gaussianity in section 2. Throughout thepresent paper, we work with models with non-vanishing primordial bispectrum; we will use the local fnl type primordial
non-Gaussianity to illustrate the calculations. The Bardeen potential Φ in the local fnl model isΦ = φ + fnl (φ2 − hφ2 i),(1)where φ is a Gaussian potential field and fnl is the nonlinear quadratic parameter. The above non-Gaussian correction isdefined at z = zCM B for this study. It has been suggested that mass weighting is important for the peculiar velocities(Scoccimarro 2004; Sheth & Zehavi 2009). We will discuss the effect of primordial non-Gaussianity on both the uniform
weighted as well as the mass weighted linear velocity PDF. In section 3 we describe the analytical model to approximate theevolution of the pairwise velocity PDF. The theoretical predictions of both the linear theory and the analytical model arecompared with measurements from N -body simulations. We conclude our findings in section 4.

2
2.1

LINEAR VELOCITY PDF
Preliminary

The linear overdensity δ(k) and the Bardeen potential Φ(k) is related by the Poisson equation
δ(k, z) = D(z)k2 M (k)Φ(k),(2)

2where M (k) = 2c T (k)/3Ωm H02 and T (k) is the matter transfer function (note we do not include the k2 factor in M (k)). The
continuity relates the linear overdensity and the peculiar velocity u(k, z):δ̇(k) + θ(k) = 0,(3)

where θ(x) ≡ ∇ · u(x) is the divergence of the velocity field. The velocity field is described solely by its divegerence since its
vorticity decays due to the expansion of the universe (for example see Bernardeau et al. 2002). Henceuj (k) = iḊ(z)kj M (k)Φ(k),(4)

2where i = −1 and the subscript j denotes the coordinate of the peculiar velocity. In this study we will be interested in therelative velocities in the parallel (k) and perpendicular (⊥a and ⊥b ) to the line of separation for two particles separated bysome distance r.
Connected moments higher than the second order vanish when the Bardeen potential is Gaussian. When the primordial
perturbation is non-Gaussian, the leading order of the non-vanishing connected moment (higher than the second order)
depends on the particular model of primordial non-Gaussianity. Most studies in the literature concern with models with a
non-vanishing bispectrum (including the local fnl and the equilateral triangle type fnl ). Some other studies investigate models
with a non-vanishing trispectrum (the gnl model, see for example Desjacques & Seljak 2010b). While this work focuses on
primordial non-Gaussianity with a leadingly non-vanishing bispectrum, it can be extended to study models where the leading
order of the non-vanishing connected moment is higher than the third order. In what follows we will use the local fnl model
to demonstrate how the non-vanishing third order connected moment modifies the pairwise velocity PDF. The calculation
can also apply to models with other types of primordial bispectrum.
We denote the peculiar velocity at position x by u(x) and the relative velocity by v(r) ≡ u(x) − u(x′ ) = u − u′ where
c 0000 RAS, MNRAS 000, 1–16

Pairwise Velocity PDF in the fnl model3x − x′ = r. We also denote the velocity difference PDF p(v; r) as the PDF of the peculiar velocity difference at two random
positions separated by r; the pairwise velocity PDF q(v; r) as the PDF of the difference of the peculiar velocities of two tested
particles separated by r. The latter PDF is the pair weighted version of the former one and they are in general not equivalent(see, for example Scoccimarro 2004). For simplicity we sometimes call the former PDF the uniform weighted PDF and thelatter one the mass/pair weighted PDF.
The numerical integration of the third order connected moments are carried out by Monte-Carlo integration using thenumerical package CUBA (Hahn 2005).
2.2

Linear Velocity Difference PDF

2.2.1

Case: fnl = 0

The linear velocity difference PDF of two random positions separated by a distance r when fnl = 0 is given by the multivariate
normal distribution:


1
1
p
p0 (v; r) =(5)
exp − v T A−1 v ,2(2π)3/2 |A|
where v = (vk , v⊥a , v⊥b ) in which vk corresponds to the relative velocity parallel to the line of separation and v⊥a and v⊥b
are the two velocity differences perpendicular to the line of separation. Here A is the covariance matrix. The above expression
simplifies since there is no correlation between different components of v. Hence the r.h.s becomes a product of three univariate
normal distributions:p0 (vk , v⊥a , v⊥b ; r) = p0 (vk ; r)p0 (v⊥a ; r)p0 (v⊥b ; r),

(6)

2
2
i = hv⊥i respectively, andwhere the variances of the univariate normal distributions are hvk2 i and hv2⊥ i ≡ hv⊥ab
j1 (kr)dk PΦ (k)k4 M 2 (k) 1 − 3j0 (kr) + 6
krZ
j1 (kr)1
hv2⊥ i =Ḋ02 dk PΦ (k)k4 M 2 (k) 1 − 3
.
3π 2
kr
hvk2 i =

1
Ḋ02
3π 2

Z

(7)
(8)

Here j0 and j1 is the spherical bessel function, Ḋ0 = dD/dt is the time derivative of the linear growth factor D at z = 0 andPΦ (k) ≈ Pφ (k) is the Bardeen potential power spectrum. Iostropy means one can transform the rectangular coordinate in theplane perpendicular to the line of separation into the polar coordinate and write equation (6) as(9)p0 (vk , v⊥a , v⊥b ; r)dvk dv⊥a dv⊥b = 2πv⊥ p0 (vk ; r)p0 (v⊥ ; r)dvk dv⊥ ,
q
2
2
2
2
2
+ v⊥
) are not equivalent.
where v⊥= v⊥
+ v⊥
. Notice that our notations hv⊥ · · ·i(= hv⊥a · · ·i = hv⊥b · · ·i) and v⊥ (= v⊥
a
b
a
b
2.2.2

Case: fnl 6= 0

When fnl 6= 0, the primordial bispectrum is non-zero and its functional form in the local type is3BΦ (k1 , k2 , k12 ) = 2fnl [P (k1 )P (k2 ) + cyclic] + O(fnl
).

(10)

Connected moments higher than the second order are non-vanishing and contribute to the velocity difference PDF. The
leadingly non-vanishing connected moments areZZ
hvk3 i12Ḋ033dk=
d3 k2 P (k1 )P (k2 )M (k1 )M (k2 )M (k12 )k1k k2k k12k [sin(k12k r) − 2 sin(k2k r)]
(11)
1
2fnl
(2π)6
cos µ2 ≥0ZZ
22
hvk v⊥i
hvk v⊥i
hvk v2⊥ i
4Ḋ03
3
a
b
d
k
d3 k2 P (k1 )P (k2 )M (k1 )M (k2 )M (k12 )[k1⊥ k2⊥ k12k sin(k12k r)
≡
=
=
1
2fnl
2fnl
2fnl
(2π)6
cos µ2 ≥0− 2k1⊥ k2k k12⊥ sin(k2k r) + 2k1⊥ k2k k12⊥ sin(k12k r) − 2k1k k2⊥ k12⊥ sin(k2k r) − 2k1⊥ k2⊥ k12k sin(k2k r)],(12)

where k12 = |k1 +k2 |, kik = ki cos µi , ki⊥ = ki sin µi cos φi , and k12k = k1k +k2k (similarly for ⊥ ). Note that connected moments
involving odd power of v⊥ vanish irrespective of the value of fnl . The non-vanishing component hvk v2⊥ i results in couplings
between velocities in the parallel and the perpendicular to the line of separation directions. As a result the linear multivariate
PDF can no longer be written as a product of three independent univariate PDFs. We use the tri-variate Edgeworth expansion
to approximate the linear uniform weighted velocity difference PDF for fnl 6= 0 (see appendix for derivation):p(vk , v⊥a , v⊥b ; fnl , r) = p0 (vk , v⊥a , v⊥b ; r)[1 + α300 h300 + α120 (h120 + h102 )],whereα300 =
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1 hvk i,6 hvk2 i3/2

α120 =

1 hvk v2⊥ i,
2 hvk2 i1/2 hv2⊥ i

(13)

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T. Y. Lam, T. Nishimichi & N. Yoshida

Figure 1. Scale dependence of parameters: α300 (thick solid black curve), α120 (thick dashed red curve),−β100 (red sketeal trianglar
symbols), β200 (thick cyan short-long-dashed curve), −γ100 (green starred symbols), γ300 (thick green dot-dashed curve), γ120 (thickblue dotted curve), −γ400 (blue pentagons), −γ220 (cyan hexagons), γ500 (thin magenta short-long-dashed curve), and γ320 thin (orange
dot-dashed curve). fnl = −100 for those parameters apply. α300 , α120 , −γ100 , and −β100 are shifted vertically by -2 for clarity.

and hijk ≡ Hi (νk )Hj (ν⊥a )Hk (ν⊥b ) is the product of Hermite polynomials of different orders. In particular,
h300 ≡ νk3 − 3νk =vk3vk− 3 2 1/2
2 3/2
hvk i
hvk i

2
h120 ≡ νk (ν⊥− 1) =avk
hvk2 i1/2

2h102 ≡ νk (ν⊥− 1) =bvk
hvk2 i1/2




2
v⊥
a
−1
2
hv⊥ i
!
2
v⊥
b
−1 .
hv2⊥ i

(14)
(15)
(16)

Note that Schmidt (2010) marginalized velocities perpendicular to the line of separation, essentially setting α120 = 0.
Figure 1 shows the scale dependence of α300 (thick solid black curve) and α120 (thick dashed red curve) for fnl = −100.
While the numerical value is small compared to unity, the effect of non-zero fnl becomes significant for big |νk | or |ν⊥ |.
Furthermore α300 and α120 are in the same order of magnitude, hence both terms have to be included in the computation of
the linear peculiar velocity difference PDF.
Figure 2 shows the ratios of the linear peculiar velocity difference PDF for fnl = 100 to the corresponding Gaussian PDF
at r = 8 h−1 Mpc. The axes labeled vll indicates the velocity in the parallel toqthe line of separation direction and vl is the22magnitude of the velocity perpendicular to the line of separation (recall vl = v⊥
+ v⊥). The upper panel shows both theabcontour and color maps over a wide range of velocity. The effect of primordial non-Gaussianity is most significant at extreme
velocities. Both the contour and the color maps show variations in the ratio for different values of v⊥ at fixed vk , indicatingthat the modification due to fnl is degenerated in the plane of (vll, vl ). While the upper panel shows that the ratio can be as
big as 4, the first order Edgeworth expansion breaks down in such rare cases (about 10-σ level) – it is evident by the negative
probability at the other end of the plot.
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Pairwise Velocity PDF in the fnl model

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Figure 2. Ratio of linear velocity difference PDF p/p0 in equation (13) for r = 8 h−1 Mpc and fnl = 100. vll indicates velocity in theparallel to the line of separation direction while vl is the magnitude of the velocity perpendicular to the line of separation. Upper panel
shows the ratio for a wide range of velocity. The lower left panel zooms into regions for around 5-σ as indicated by the outer axes. The
lower right panel shows the same regions as the lower left panel, but explicitly set the parameter α120 = 0 as previous study did.

The lower panels of Figure 2 show the color maps of the same PDF ratio on a smaller velocity range to increase thedynamic range of the effect of fnl on the linear velocity difference PDF. The left panel shows the modification obtained inthis study (the correction term in equation (13)) while the right panel shows the result for setting hvk v2⊥ i = 0. The outer axesof the lower panels show the σ-level in corresponding directions. At 5-σ level (vll direction) the change is about 30%, and thebigger the σ-level in the perpendicular direction the greater the modification is. The linear theory suggests that for fnl > 0
the infalling probability (negative velocity) is enhanced while the outgoing probability (positive velocity) is decreased. This
trend applies to other scales we looked at (from 4 to 100 h−1 Mpc). While it is generally believed that linear theory applies
for large scales, we will show in the next section that linear theory fails to describe the change in the velocity PDF due to fnl
even at separation as big as 50 h−1 Mpc.

2.3

Linear Pairwise Velocity PDF

It is widely accepted that, while the linear theory prediction is a good approximation for large scales (& 20 h−1 Mpc), thelinear theory velocity correlations of massive halos (in the parallel to the line of separation direction) is not consistent withc 0000 RAS, MNRAS 000, 1–16

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T. Y. Lam, T. Nishimichi & N. YoshidaN -body measurements (Croft & Efstathiou 1995). Sheth & Zehavi (2009) pointed out that the velocity correlations of biased
tracers, in both parallel and perpendicular to the line of separation directions, can in fact be reasonably described by linear
theory when proper pair-weighting is included. While our current study focuses on the difference of the peculiar velocities of
unbiased tracer, it is useful to examine whether the pair weighting would improve the linear theory prediction. It is reasonable
to include this mass weighting as the pairwise velocity PDF measured in N -body simulations is obtained by counting therelative velocities of simulated particles. As a result regions with more particles (overdense regions) have a bigger weight thanregions with less particles (underdense regions).
Case: fnl = 0

2.3.1

Scoccimarro (2004) discussed how to generalize the linear velocity difference PDF when fnl = 0 to include the mass weighting.
The resulting linear pairwise velocity PDF q0 is related to the associated no weighting velocity difference PDF p0 by[1 + ξ(r)]q0 (νk , ν⊥a , ν⊥b ; r)
= 1 + ξ(r) + h100 β100 + h200 β200 ,p0 (νk , ν⊥a , ν⊥b ; r)(17)where δ ≡ δ(x), δ ′ ≡ δ(x + r), ξ ≡ ξ(r) = hδδ ′ i is the matter correlation function at separation r, andhvk δihvk δ ′ i
hvk δihvk δ ′ i,(18)+ 2 1/2 ,β200 =
2 1/2
hvk2 i
hvk i
hvk i
Z
1
Ḋ0 dkPΦ (k)k4 M 2 (k)[kj1 (kr)].
(19)
hvk δi = hvk δ ′ i =2π 2
The r.h.s of equation (17) does not depend v⊥ , hence q0 can still be written as a product of three univariate PDFs – thecorrection term on the r.h.s of equation (17) only modifies the univariate PDF of vk . The scale dependence of the parameters−β100 (red sketeal triangular symbols) and β200 (thick cyan short-long-dashed curve) are shown in Figure 1: their magnitudes
are big compared to others shown in the same figure since they are ’Gaussian parameters’ that do not depend on fnl .
β100 =

Case: fnl 6= 0

2.3.2

When fnl 6= 0 additional terms contribute to the mass weighted linear velocity difference PDF. To the first order of fnl theexpression is (see derivation in appendix)[1 + ξ(r)]q(νk , ν⊥a , ν⊥b ; fnl , r)
= 1 + ξ(r) + h100 γ100 + h200 β200 + h300 γ300 + (h120 + h102 )γ120
p0 (νk , ν⊥a , ν⊥b ; r)
+ h400 γ400 + (h220 + h202 )γ220 + h500 γ500 + (h320 + h302 )γ320 ,

(20)

where
γ100 = β100 +hvk δδ ′ i
hvk2 i1/2

β100
2γ320 = α120 β200
γ400 = α300

2 ′γ300 = α300 (1 + ξ(r)) +β100 hvk δ i
2 hvk2 iγ120 = α120 (1 + ξ(r)) +β100 hv2⊥ δ ′ i
2 hv2⊥ iγ220 = α120 β100

γ500 = α300 β200

h400 = νk4 − 6νk2 + 3h500 = νk5 − 10νk3 + 15νk ,and for the local fnl type primordial non-GaussianityZZ
hvk δδ ′ i
4Ḋ0
3
2
d
k
=−
d3 k2 P (k1 )P (k2 )M (k1 )M (k2 )M (k12 )[k1k k22 k12
sin(k12k r) + k12 k22 k12k sin(k2k r)
1
2fnl
(2π)6
cos µ2 ≥0

2− k1k k22 k12
sin(k2k r)]
Z
Z
2 ′
2
hvk δ i
2Ḋ0
3
2
d
k
d3 k2 P (k1 )P (k2 )M (k1 )M (k2 )M (k12 )[2k1k k22 k12k cos(k2k r) − k1k k2k k12
cos(k12k r)
=
1
2fnl
(2π)6
cos µ2 ≥0

(21)

2
2
+ 2k1k k2k k12
cos(k2k r) − 2k1k k22 k12k cos(k1k r) − 2k1k k22 k12k cos(k12k r) + 2k1k k22 k12k − k1k k2k k12
]
(22)
Z
Z
2 ′
2
hv⊥ δ i2Ḋ02
d3 k 1
d3 k2 P (k1 )P (k2 )M (k1 )M (k2 )M (k12 )[2k1⊥ k22 k12⊥ cos(k2k r) − k1⊥ k2⊥ k12
cos(k12k r)
=
2fnl
(2π)6
cos µ2 ≥0
2
2+ 2k1⊥ k2⊥ k12
cos(k2k r) − 2k1⊥ k22 k12⊥ cos(k1k r) − 2k1⊥ k22 k12⊥ cos(k12k r) + 2k1k k22 k12k − k1k k2k k12
].

(23)

The scale dependences of γijk are shown in figure 1. As in the case of uniform weighted linear PDF, the linear pairwise velocity
PDF can no longer be written as a product of three independent PDFs when fnl 6= 0 due to the non-vanishing terms hvk v2⊥ iand hv2⊥ δ ′ i. In addition, when one sets δ = δ ′ = 0, the above expression recovers the uniform weighted linear PDF for fnl 6= 0
(equation (13)).
Figure 3 shows the modification factor of the linear pairwise velocity PDF due to fnl (q/q0 in equations (17) and (20)). As
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Pairwise Velocity PDF in the fnl model

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4

Figure 3. Similar plot to figure 2 but for the ratio of the linear pairwise velocity PDF (q/q0 in equations (17) and (20)). The lower rightpanel sets α120 = γ220 = γ320 = hv2⊥ δ ′ i = 0.

in Figure 2 the upper panel shows the ratio in a wide range of velocity, including regions with negative probability indicating
the breakdown of the first order Edgeworth expansion approximation. The lower left panel enlarges the color map on regions
about 5-σ level and the outer axes label the σ-level. The lower right panel shows the ratio if one explicitly sets all third ordermoments involving v⊥ to zero (i.e. α120 = γ220 = γ320 = hv2⊥ δ ′ i = 0). The effect of fnl in the mass weighted linear pairwise
PDF has the same trend as the uniform weighted PDF: the effect of fnl is degenerated in the parallel and the perpendicular
to the line of separation directions; and the general trend observed in uniform weighted linear PDF still applies – for fnl > 0
the probability of finding infalling pairs increases while the probability of finding pairs moving apart decreases. The inclusionof the mass weighted quantites γijk strengthens the effect of fnl and it is most significant in the decrement in probability ofoutgoing pairs.

3

EVOLVED VELOCITY PDF

Given the linear velocity PDF we now describe a model for the evolution of the PDF. We adopt the Zeldovich Approximation
which assumes the comoving velocity remains unchanged. At some initial redshift zi , if two particles separated by ri have aiiiirelative velocity (vki , v⊥
, v⊥). Then at a later redshift z = z0 (≪ zi ) the distances traveled are D0 /Ḋi (vki , v⊥
, v⊥
) (noteaabbc 0000 RAS, MNRAS 000, 1–16

8

T. Y. Lam, T. Nishimichi & N. Yoshidathat it assumes D0 ≫ Di ). Hence the separation of the two particles becomes

2 
2
D0 i
D0
i 2
i 2
r 2 = ri +
vk
+
+ v⊥
(v⊥
).
a
b
Ḋi
Ḋi
The evolved relative velocities (with respect to the updated position) are:!
i
D0 2
Ḋ0 ri vk+ 2 vi
vk =
r
Ḋi
Ḋi
2

Ḋ0 i
2
2
v
− vk2 ,+v=
|v⊥ |2 = v⊥
⊥b
a
Ḋi
2

2

2

(24)

(25)
(26)

2

i
i
+ v⊥
where v i = vki + v⊥
. The evolved velocity difference PDF is therefore
a
b
Z
ri2
i
i
i
i
dv⊥
p(Vk , V⊥ ; R) = dri dvki dv⊥, v⊥; r)δD (r − R)δD (vk − Vk )δD (v⊥ − V⊥ ),p(vki , v⊥
a
a
b
b
R2

(27)

(see, for example, Seto & Yokoyama 1998) where the dirac-delta functions use equations (24), (25), and (26) to map the initialquantities to the evolved one. In the following we apply the above formula to compute the evolved velocity difference PDF
from equations (6) and (13) and compare the results to measurements from N -body simulations. We set zi and z0 to be 1100
and 0.5 respectively, justifying the approximation D0 ≫ Di .

3.1

Comparisons to N -body measurements

We measured the pairwise velocity PDF from a set of N -body simulations described in Nishimichi et al. (2009). The simulation
adopts the WMAP 5-years ΛCDM best fit parameters (Ωm , ΩΛ , Ωb , h, σ8 , ns ) = (0.279, 0.721, 0.046, 0.701, 0.817, 0.96). The
simulation was performed in a box of 2000 h−1 Mpc on a side, containing 5123 particles whose mass is 4.6 × 1012 h−1 M⊙ . The
measurements were made from the simulation output at z = 0.5. To account for the discrete nature of N -body measurements,
separation r in the following refers to two particles having a separation between (r − 2, r + 2) in the unit of h−1 Mpc. For
the theoretical model we apply the same separation selection in the dirac-delta function of r. The comparisons of the velocity
PDF of vk and v⊥ are presented in the next two subsections. We will then discuss the implications of the comparisons in thenext section.

3.1.1

Parallel to the line of separation: p(vk ; r, fnl )Figure 4 shows the PDF comparisons of the various analytical predictions to the N -body measurements for the relative velocity
parallel to the line of separation for 4 different separations (4, 8, 12, 50 h−1 Mpc). The upper panel in each subfigure shows
the PDF profile for fnl = 0: solid symbols are the N -body measurements, cyan dotted curves are the analytical predictions ofequation (27) (marginalized over v⊥ ), green dot-long dashed (mass weighted) and magenta long dashed (uniform weighted)
curves are the linear theory predictions respectively.
The predictions from our analytical evolution model give good matches to the N -body measurements at all scales weinvestigated, except at large outgoing velocities (i.e. vk ≫ 0). The agreements with the N -body measurements are better insmall separations (4 and 8 h−1 Mpc) than large separation (12 h−1 Mpc). The analytical predictions of our model are able todescribe the high infalling velocity regimes (vk < −1000). This is very encouraging since one may expect that high velocity
regions are not described by our simple model. The analytical model also predicts knee-shape transitions at around 300, 600,
and 900km/s for r = 4, 8, 12 h−1 Mpc respectively. Similar but less significant knee-shape changes can be seen from the
N -body measurements around the corresponding velocities. In the high outgoing velocity regions the match is not as good as
the infalling regimes. We believe the disagreement in the outgoing velocity regime as compared to the infalling regime is due
to non-linear evolution of velocity that is not described by our current model – originally infalling pairs eventually become
outgoing (see equation (25)) and non-linear interactions between particles in close contact are not described by the Zeldovich
Approximation.
On the other hand, the linear theory predictions do not fare as well as the analytical model’s. At small separations thepredictions of the linear velocity difference PDF do not match the N -body measurements – the magneta curves miss both theextrema and the peaks of the PDF at separation r = 4, 8, and 12 h−1 Mpc. This is consistent with previous studies that thelinear theory uniform weighted velocity PDF does not agree with the velocity correlation in the parallel to the line of separation
direction. On the contrary the mass weighted linear predictions provide reasonable matches to the N -body measurements
near the peaks of the PDF at different separations. It confirms the finding of Sheth & Zehavi (2009): the velocity correlation
can be described by the linear theory when the mass weighting is taken into account. The matching of the peaks of the PDF
guarantees the velocity correlations, which is equivalent to the expected value of the pairwise velocity PDF, would roughly
agree. However the mass weighted linear predictions does not match the N -body measurements when |vk | is of the order ofc 0000 RAS, MNRAS 000, 1–16

Pairwise Velocity PDF in the fnl model(a) r=4 Mpc h−1(b) r=8 Mpc h−1(c) r=12 Mpc h−1 .

(d) r=50 Mpc h−1

9Figure 4. Pairwise velocity (parallel to line of separation) PDF at different separations. Negative velocity corresponds to particles
infalling while positive velocity indicates particles moving away from each other. Upper panels show the pairwise velocity PDF when
fnl = 0: solid symbols are measurements from the N -body simulation, cyan dotted curves are the velocity difference PDF from theevolution model based on the Zeldovich approximation (equation (27)), green dot-long dashed curves are the linear pairwise velocity (q0in equation (17)), and magenta long dashed curves are the linear velocity difference PDF (equation (6)). Lower panels show the ratiosof the PDF for fnl ± 100 to the associated PDF for fnl = 0: crosses (fnl = 100) and solid squares (fnl = −100), cyan dotted curves(fnl = 100) and blue dot-short dashed curves (fnl = −100) are predictions of equation (27), green dot-long dashed curves are the ratios
for the linear pairwise PDF (fnl = 100), and magenta long dashed curves are the ratios for the linear velocity difference PDF (fnl = 100).
Red short-long-dashed curves show the prediction from equation (27), neglecting third order moments involving velocity perpendicular
to line of separation (see the lower right panel of figure 2).

a few hundreds km/s. The disagreement is getting worse as the separation decreases. At large separation (50 h−1 Mpc), thetwo linear theory predictions are very similar and they describe the N -body measurements well. They are very closed to theprediction of the analytical model and the match to the measurements is only slightly worse at high velocity regimes.
The lower panels of figure 4 show the ratios (substracted by unity) of the pairwise velocity PDFs for fnl = ±100 to theassociated PDFs for fnl = 0 at different separations. Measurements from the N -body simulations are represented by crosses(fnl = 100) and squares (fnl = −100). Other curves in the lower panels show the PDF ratios of different analytical predictions:c 0000 RAS, MNRAS 000, 1–16

10

T. Y. Lam, T. Nishimichi & N. Yoshida

cyan dotted (fnl = 100) and blue dot-short dashed (fnl = −100) curves are the predictions of the evolution model based on
the Zeldovich Approximation (equation (27)); red short-long-dashed curves show the similar predictions for fnl = 100 but
setting hvk v2⊥ i = 0 (see the lower right panel of figure 2 for the corresponding linear theory comparison at r = 8 h−1 Mpc).
The other two curves are the linear theory predicted ratios for fnl = 100: the magneta long dashed curves are the uniform
weighted velocity predictions and the green dot-long dashed curves are the mass weighted velocity predicted ratios.
For the range of separations we investigated the change in the pairwise velocity PDF due to fnl = ±100 is at most 5%
in the infalling velocity regime and 10% in the outgoing velocity regime. In contrary to the linear theory predictions (seefigures 2 and 3, and the magenta long dashed as well as the green dot-long dashed curves), the effect of positive fnl enhances
the probability of having pairs in both the infalling and the outgoing high velocity ends. This is true for all the separations
we look at, from 4 to 50 h−1 Mpc – scales in which the linear theory is believed to be valid. We also check using a simulation
with a smaller box (1 h−1 Gpc on a side) to estimate the box size effect: shrinking the box volume to one-eighth of its sizechanges the PDF ratios by at most 2% at vk = 2000 km/s.
The predictions of our evolution model, regardless of whether one explicitly set hvk v2⊥ i = 0 , match the N -body measurements reasonably well when r = 4 h−1 Mpc. The predicted ratios from the analytic model match the measurements within
2% across the velocity range we look at. It is remarkable to have such agreement at velocity as high as ∼ 2000km/s. The
predicted ratios also match the transition of the PDF from the infalling to the outgoing regime: for fnl = 100 it gradually
decreases from positive to negative, then changes direction and become positive again. The same transition is also observedat larger separations and the model’s predictions are able to match the N -body measurements. In contrast the ratios of thelinear theory predictions (both the uniform and the mass weighted) fail to match the N -body measurements in both infallingand outgoing velocity regimes. While the linear theory predicted ratios still have the same trend as the N -body measurementsin the infalling velocity regime (but the differences between the predictions and the measurements are large), the predictedratios from linear theory are opposite to the measurements in the outgoing velocity regime (the same is also observed at larger
separations). Hence the linear theory cannot be used to predict the effect of fnl on the pairwise velocity PDF.
At larger separations the agreements in the ratios of the PDF between our analytical model predictions and the N body measurements are less impressive: while the predicted ratios still agree well with the measurements in the infalling
velocity regime, the differences in the outgoing velocity regime are getting bigger as the separation increases. Nonethelessthe predictions still qualitatively match the measurements at r = 8 and 12 h−1 Mpc. At r = 50 h−1 Mpc, on the other hand,the analytical predictions converge to the linear theory predictions and show opposite trends to the measurements. In theoutgoing velocity region, there are some interesting transitions in the analytical model predictions: at r = 12 h−1 Mpc and
fnl = 100, the predicted ratios (the cyan as well as red curves) first show a decrement at small vk . They then turn aroundnear vk = 800km/s and keep increasing for larger vk . Such transitions are not seen in the N -body measurements.
We will leave the discussion on the implications for the discrepancies between the linear theory predictions and the
N -body measurements at all separations as well as the mismatch in the PDF ratios between our analytical models and themeasurements at large separations in the next section.
3.1.2

Perpendicular to the line of separation: p(v⊥ ; r, fnl )Figure 5 shows the comparisons of the PDFs of the pairwise velocity perpendicular to the line of separation for different
separations. The upper panels show the PDF at z = 0.5 for fnl = 0 and the lower panels show the ratios of the PDF for
fnl = ±100 to the associated PDF for fnl = 0. The symbol and curve labelings are the same as in figure 4, but we do notshow the PDF ratio predictions of the linear theory in the lower panels since the linear theory predicts no effect of fnl in themarginalized PDF, that is p(v⊥ ; r, fnl )/p(v⊥ ; r, fnl = 0) = 1.
The predictions of the PDF profile from our analytical model (equation (27)) match the N -body measurements reasonably
well, except around v⊥ = 0 at r = 50 h−1 Mpc. The predictions agree with the measurements in a wide range of velocity(200 − 1500km/s), as well as the dips near v⊥ = 0. At higher v⊥ the model predictions always underestimate the PDF. The
linear theory predictions, on the other hand, fail to match the measured PDF for r = 4, 8, and 12 h−1 Mpc. At r = 50 h−1 Mpc
the linear theory prediction agrees with the N -body measurements for v⊥ = 300 − 1200km/s, but it fails to predict the dip
near v⊥ = 0. Note that there is no difference between the mass weighted and the uniform weighted linear theory predictions
since the correction term due to mass weighting does not depend on v⊥ (equation (17)).
The effect of primordial non-Gaussianity in the pairwise velocity PDF in the perpendicular to the line of separation
direction is not as strong as the signature in the parallel to the line of separation direction: the change in the PDF is top at
7% at v⊥ = 2000km/s. The ratio first shows a decrement (increment) at small v⊥ for fnl = 100 (−100). It then graduallyswitches direction and at high v⊥ show an increment (decrement). This gradual switch from decrement to increment ismost significant at small separation. Our analytical model is able to predict the change in the PDF due to primordial nonGaussianity qualitatively. Its prediction matches the gradual change of the PDF ratios (from decrement to enhancement for
fnl = 100, opposite for fnl = −100) and provides a rough approximation on the velocity where the crossing across unity
occurs. However the predictions always underestimate the PDF ratios in low v⊥ regime. At higher v⊥ the model prediction on
c 0000 RAS, MNRAS 000, 1–16

Pairwise Velocity PDF in the fnl model(a) r=4 Mpc h−1(b) r=8 Mpc h−1(c) r=12 Mpc h−1(d) r=50 Mpc h−1

11Figure 5. Pairwise velocity (perpendicular to line of separation) PDF at different separations. Upper panels show the PDF profile when
fnl = 0: solid symbols are measurements from the N -body simulation, cyan dotted curves are the prediction from the evolution model
based on the Zeldovich Approximation (equation (27)), green dot-long dashed curves are the linear pairwise velocity (q0 in equation (17)),and magenta long dashed curves are the linear velocity difference PDF (equation (6)). Lower panels show the ratios of the PDF when
fnl ± 100 to the associated PDF for fnl = 0: crosses (fnl = 100) and solid squares (fnl = −100) are measurements from N -bodysimulation, cyan dotted curves (fnl = 100) and blue dot-short dashed curves (fnl = −100) are predictions of equation (27). Red shortlong-dashed curves show the prediction from equation (27), neglecting third order moments involving velocity perpendicular to line ofseparation (see the lower right panel of figure 2)

the PDF ratio matches the N -body measurements very well at r = 8 as well as 12 h−1 Mpc; the predicted ratio overestimatesthe change at r = 4 h−1 Mpc but underestimates at r = 50 h−1 Mpc. The analytical model predicts the effect of fnl becomes
weaker at some fixed v⊥ for larger r, the measurements from the N -body simulations, on the contrary, remain roughly thesame at various separations. The measured PDF ratios at high v⊥ only depends on the scale of separation very weakly. In
addition there are spikes near v⊥ = 0 in the theoretical predicted ratios for r = 12 and 50 h−1 Mpc, but these are not seen in
the N -body simulations.
The linear theory predictions fail to match the N -body measurements at all separations – it predicts no change in thec 0000 RAS, MNRAS 000, 1–16

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T. Y. Lam, T. Nishimichi & N. Yoshida

pairwise velocity PDF in the perpendicular to the line of separation direction. The N -body measurements, on the other hand,
has several percent-level change in the PDF throughout the velocity range we look at.

4

DISCUSSIONS

We have studied how primoridal non-Gaussianity affects the probability density function of the pairwise velocity. Our study
complements earlier studies on the signatures of primordial non-Gaussianity on other large-scale structure probes which utilize
the modification in the density field (for example the PDF of density field, halo/void abundances, scale dependent halo bias).
We have shown that primordial non-Gaussianity models with a non-vanishing bispectrum result in two non-vanishing thirdorder connected moments of the peculiar velocity (hvk3 i and hvk v2⊥ i). We adopt the local fnl type primordial non-Gaussianity
as an example to compute the modification in the linear velocity difference PDF. Following suggestions by earlier studies,
we investigate the effect of fnl on the linear pairwise velocity PDF. Primordial non-Gaussianity induces correlations between
velocities in the parallel and the perpendicular to the line of separation directions. Hence both the uniform weighted and themass weighted linear velocity PDF can no longer be written as a product of the corresponding univariate PDFs. Futhermore
the change in the linear velocity PDF is degenerated in the (vk , v⊥ ) plane.
Next we have developed an analytical model to describe the evolution of the velocity PDF. It employs the Zeldovich
Approximation which assumes the comoving velocity remains unchanged. We then compared the pairwise velocity PDF
measured from a N -body simulation with the predictions from our analytical model at various separations, ranging from 4
to 50 h−1 Mpc. Our model predictions agree well with the PDF profiles measured from the N -body simulation at various
separations when fnl = 0 in both the parallel and in the perpendicular to the line of separation directions.
Linear theory fails in matching the N -body measurements: both the linear velocity difference PDF as well as the linear
pairwise velocity PDF fail to predict the PDF profiles and the change in the PDF due to a non-zero fnl . The linear theory
prediction does not take into account the movement of particles: pairs separated by r having a relative velocity (vk , v⊥a , v⊥b )must have had different separations at earlier redshift. The relative velocities at earlier redshifts also change in general dueto the difference in the particles’ positions. In other words linear theoy is a good approximation only when the displacementof the particles is small compared to the separation between them. For example the displacement associated with a relative
velocity v = 1000 km/s is approximately 10 h−1 Mpc at z = 0.5 if the velocity remains unchanged. Therefore, only for large
enough separations (r ≫ 10h−1 Mpc) the linear theory can approximate the PDF. The initial separation and the initial
relative velocity determine the evolved separation and the pairwise velocity – equations (24) to (26) give an example of suchian evolution model using the Zeldovich approximation. In general the mapping from the initial parameters {ri , vki , v⊥
} to theiiievolved parameters {r, vk , v⊥ } is non-linear and more than one set of {r , vk , v⊥ } would evolve to {r, vk , v⊥ }. This analyticalmodel provides results that match the measurements much better than the linear theory.
Including evolution is also important in explaining the change in the PDF due to fnl . The ratios of the pairwise velocityi
PDF in the parallel to the line of separation direction can be used to reveal the distribution of pairs with different {ri , vki , v⊥}
having evolved to the same {r, vk , v⊥ }. Recall that the linear theory predicts enhancement in the probability occurs in highvelocity infalling pairs and low velocity outgoing pairs when fnl > 0. On the other hand, the N -body measurements always
have increment in the probability for outgoing pairs. The analytical model provides an algorithm to explain this discrepancy:
pairs initially infalling becomes outgoing when they cross each other on the projection on the line of separation. As a result
outgoing pairs would be either originally infalling or outgoing pairs. The converse is not true because infalling pairs can only
come from pairs that approach each other at earlier redshift. The originally low velocity outgoing pairs, having a higher
probability for fnl = 100, by itself cannot explain the observation from N -body measurements as the change in the velocity
2
difference PDF associated with these pairs is relative small (recall that the change in the PDF goes as vk3 − 3vk and v⊥− 1)
and cannot account for the level of change in the PDF measured in the simulations. As a result the PDF ratio in the outgoingvelocity regime has to be dominated by original infalling pairs.
Our analytical model provides an approximation for the evolution of the velocity difference PDF and it successfully
predicts the modification in the velocity PDF due to fnl qualitatively. The Zeldovich Approximation, on which the analyticalmodel based, does not take into account several important features – it assumes the comoving velocity remains unchanged.
This is a good approximation when the separation of the pair in consideration is big. This assumption may not hold when thepair crosses on the projection along the line of separation. Non-linear model is required to describe the evolution of the relative
velocity in this cases. As a result the velocity PDF in the regime where cross-over dominates may not be well described by
our analytical model.
Another area our analytical model does not describe is the virial motion of particles residing inside dark matter halos.
While it is not expected to have pairs residing within the same dark matter halo for the separation we considered in thisstudy, virial motions would certainly affect the distribution of the pairwise velocity.
Notice that our analytical model only describes the evolution of the velocity difference PDF – it is the evolved uniform
weighted velocity PDF. In practice it is not what we measured from the N -body simulation – it is a (evolved) pair weighted
c 0000 RAS, MNRAS 000, 1–16

Pairwise Velocity PDF in the fnl model

13

velocity PDF. The comparisons with N -body measurements show that the evolved velocity difference PDF matches themeasurements better than the linear pairwise velocity PDF. Modeling of the non-linear pairwise velocity PDF requires thecorrelation between the non-linear density contrast and the evolved velocity field. These are not described in the our modeland will be left for future study.
It is quite unexpected that the agreement of the analytical model with the N -body measurement is better at small
separation than at large separation. One speculation for that is the simulation measurement is affected by the finite box
size effect for big r – we did check the effect of finite box size by measuring the pairwise velocity PDF from a simulation
ran in a box with 1 h−1 Gpc on a side. The resulting PDF at r = 50 h−1 Mpc agrees qualitatively with the corresponding
PDF in the main text. Hence we believe the disagreement in the PDF ratio at r = 50 h−1 Mpc is not due to the effect offinite box size. Another possibility is the discrepancy is due to the non-linear interactions – here we give two approaches to
explain qualitatively how the non-linear interactions affect the velocity PDF. Both are related to the formation of non-linear
structures and is not described by the Zeldovich Approximation.
The Zeldovich Approximation assumes the comoving velocity is constant throughout the evolution and it is not a good
approximation when non-linear structures (pancake, filament, and halos) form. If the Zeldovich Approximation were thecorrect model that completely described the evolution, the positive velocity in the evolved velocity difference PDF would
be dominated by originally infalling pairs only when vk & v0 , where v0 corresponds to the velocity that has a displacement
approximately equal to the separation r. This scale also associates with the knee-shape transition in the PDF profiles in theupper panels of figure 4. For vk < v0 the Zeldovich Approximation predicts originally outgoing pairs should dominate the
PDF. Measurements from N -body simulation, on the other hand, show no such transition. The knee-shape transition in the
PDF profiles (upper panels) is also much weaker compared to the analytical predictions. Non-linear evolution models, such as
the adhesion model, may be able to explain the difference between the measured and the predicted PDF – while the Zeldovich
approximation predicts the velocity PDF is dominated by originally infalling pairs only when vk > v0 , the frictional dragging
term in the adhesion model would damp the velocity, making v0 < vk′ → vk < v0 . As a result the velocity PDF would bedominated by originally infalling pairs at a smaller vk compared to the Zeldovich Approximation prediction. Notice that thiseffect will be most significant for big r as the dragging term is proportional to v.
The halo model provides another explanation for this discrepancy: in the halo model all particles are residing in halos,each of them acquires its peculiar velocity from two contributions – the virial motion inside the parent halo as well as the bulk
movement of the parent halo (Cooray & Sheth 2002; Sheth & Diaferio 2001; Sheth et al. 2001). Hence the relative velocity
between pairs residing in different halos (this corresponds to the separations we studied in the current study) can be separated
into two parts: the randomly-oriented velocity difference due to the corresponding virial motion and the relative velocity ofthe two parent halos. The dispersion of the virial motion depends on the mass of the parent halo: more massive halos have alarger dispersion; the bulk velocity of the parent halo depends only weakly on mass. The bulk velocity can be approximatedby the linear theory (Cooray & Sheth 2002). At relatively small separations, the two contributions are comparable; at big
separations, on the other hand, the two-point correlation function is weak and the velocity is dominated by the virial motion
term. It is a well-known fact that there are more massive halos for models with fnl > 0 (see, for example, Lam et al. 2009),
hence the velocity dispersion is bigger when fnl > 0. This effect is most significant in the pairwise velocity PDF for big r sinceat such separation the dominant velocity difference is the random-oriented virial velocity difference. The net result is having
a higher probability when |vk | is big but lower probability when |vk | is small. Our current analytic model does not include
the virial motion and as a result it is not able to describe the change in the velocity PDF for large r. Note that this picture
naturally predicts the discrepancy should first show in the outgoing velocity regime – clustered halos attract each other andhence the relative bulk velocity is always infalling when projected on the line of separation of the pairs in consideration.
Figure 6 shows the change in the PDF of peculiar velocity for models with fnl 6= 0. The increase in velocity dispersion and thechange in PDF at high v when fnl = 100 supports this halo model picture in explaining the discrepancy at large separation
due to the increase in the magnitude of the virial motion in fnl (> 0) models.
It is out of the scope of the current study to include these non-linear effect – this will be left for future study. Our current
analytic model is able to predict the modification in the pairwise velocity PDF within a few percent level at quasi-linear
regime (from 4 − 12 h−1 Mpc).
While this study focuses on the pairwise velocity of PDF of dark matter, we are in the process to extend the current
work to study the effect of primordial non-Gaussianity on the pairwise velocity of biased tracers. We are also extending ourstudy to examine the effect of primordial non-Gaussianity on the redshift space distortion. The N -body measurements of the
pairwise velocity PDF show clearly both the parallel and the perpendicular to the line of separation directions have primordial
non-Gaussianity imprints. The effect can be seen in separation as large as r = 50 h−1 Mpc. Our analytical model improves
earlier work (Lam et al. 2010; Schmidt 2010) – it includes the change in the pairwise velocity PDF in both the parallel andthe perpendicular directions in the analysis as well as the evolution of the pairwise velocity PDF. As a result it provides amuch better match to the N -body measurements compared to the linear theory. This enables our analytical model to improvethe estimation of the signature of primordial non-Gaussianity in the redshift space distortion.
c 0000 RAS, MNRAS 000, 1–16

14

T. Y. Lam, T. Nishimichi & N. YoshidaP(|u|; fnl) / Pg(|u|) − 1

Pg(|u|)

10−2
10−3
10−4
10−5
0.04

fnl = 100

0.02
0.00
−0.02

fnl = −100

−0.04
0

500

1000
|u| [km/s]

1500

2000

Figure 6. The 1-point PDF of peculiar velocity: the upper panel shows the PDF profile of the peculiar velocity when fnl = 0. The
open circles show the measurement from N -body simulation and the solid curve shows the Maxwellian distribution with σ equal tothe velocity dispersion measured from the simulation. The lower panel shows the ratio of the PDF for fnl = ±100 to fnl = 0: upward
pointing triangles are the ratio for fnl = 100; downward pointing for fnl = −100. Particles with large velocities (> 1000km/s) are mostly
populated in massive halos.

ACKNOWLEDGEMENTS
TYL would like to thank Ravi Sheth, Robert Smith, and Vincent Desjacques for the discussion on peculiar velocity field. This
work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. TYL is also
supported by a kakenhi grant (Grant-in-Aid for Young Scientists (B) – 22740149). T. N. is supported by a Grant-in-Aid forJapan Society for the Promotion of Science (JSPS) Fellows.

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c 0000 RAS, MNRAS 000, 1–16

16

T. Y. Lam, T. Nishimichi & N. Yoshida

APPENDIX A: DERIVATION OF MASS WEIGHTED LINEAR PAIRWISE VELOCITY PDF
This appendix derives the expression of the linear pairwise velocity PDF for fnl 6= 0 (equation (20)). The following derivation
also applies to the derivations of the linear velocity difference PDF when fnl 6= 0 (equation (13), by setting δ = δ ′ = 0 in thefollowing derivation) and the linear pairwise velocity PDF when fnl = 0 (equation (17)).
The first step is to generalize the expression for the line-of-sight pairwise velocity generating function M (equation (8)in Scoccimarro (2004)) to the 3d generating function of pairwise velocity:Z(λ; r) ≡ [1 + ξ(r)]M(λ; r) = hexp(λ · v)[1 + δ(x)][1 + δ(x′ )]i,

(A1)

′

where λ · v = λk vk + λ⊥a v⊥a + λ⊥b v⊥b and x = x + r. The characteristics function can be computed using the cumulants(see discussion in Scoccimarro (2004)):Z(iλ; r) = exphexp(iλ · v)ic [1 + hexp(iλ · v)(δ + δ ′ )ic + hexp(iλ · v)δic hexp(iλ · v)δic + hexp(iλ · v)δδ ′ ic ]#"2
2
X hvj2 i
hvk3 ihvk v⊥ii
hvk v⊥
22
3
2ab(iλj ) +(iλk ) +(iλk )(iλ⊥a ) +(iλk )(iλ⊥b ) + . . .
= exp
2
3!
2
2
j
(

X
hvj2 (δ + δ ′ )i× 1 + ξ(r) + hvk δδ ′ i(iλk ) +hvj (δ + δ ′ )i(iλj ) +(iλj )2 + . . .
2
j
"
#"
#
)
X
X
hvj2 δi
hvj2 δ ′ i
22′+
hvj δi(iλj ) +(iλj ) + . . .
(iλj ) + . . . + . . . ,hvj δ i(iλj ) +
2
2
j
j

(A2)

(A3)

where δ ≡ δ(x), δ ′ ≡ δ(x′ ), and the summation over j denotes summing over all three directions {k, ⊥a , ⊥b }. The first termin the second equity is the linear velocity difference characteristic function while the second term is associated with the mass
weighting. The characteristic function for the linear velocity difference for fnl = 0,"#
X hvj2 i
2F0 (iλ; r) = exp(iλj ) ,(A4)2
jwhose Fourier transofrm is the linear velocity difference PDF when fnl = 0 (equation (6)). Hence we can factorize F0 from Z
and define G
Z(iλ; r) ≡ G(iλ; r)F0 (iλ; r)(A5)

So far the expression is exact. Now we take the approximation that only terms up to the first order in fnl is included andnotice that hv⊥a δi = hv⊥b δi = 0. Hence#" 3
hvk δihvk2 δ ′ i
hvk δ ′ ihvk2 δi
hvk i
′
′ 
′
2
(iλk )3(1 + ξ(r)) ++G(iλ; r) = 1 + ξ(r) + hvk δi + hvk δ i + hvk δδ i (iλk ) + hvk δihvk δ i(iλk ) +
3!
2
2
+

X

hvk3 ihvk3 ihvk δi + hvk δ ′ i (iλk )4 +hvk δihvk δ ′ i(iλk )5 +Γ(iλ⊥α ),
3!
3!

(A6)

α=a,b

where

Γ(iλ⊥α ) =




2
2
2
2

δ′ i
hvk δihv⊥i
δi
i
hvk v⊥
hvk δ ′ ihv⊥
hvk v⊥αααα++(1 + ξ(r)) (iλk )(iλ⊥α )2 +hvk δi + hvk δ ′ i (iλk )2 (iλ⊥α )2
2
2
2
2

2
hvk v⊥iα(A7)hvk δihvk δ ′ i(iλk )3 (iλ⊥α )2 .
2
To get the mass weighted linear PDF we need to do a Fourier transform on Z. While the expression seems very complicated,
the Fourier transform can be done analytically due to the fact that the Fourier transform of G is actually a derivative operator
acting on the uniform weighted linear PDF. Recall that the uniform weighted linear PDF is a product of three univariate
PDFs, hence the derivative operation can be done easily and the resulting mass weighted linear pairwise velocity PDF is
equation (20).

+c 0000 RAS, MNRAS 000, 1–16