The pairwise velocity probability density function in models with local primordial non-Gaussianity

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by  Tsz Yan Lam

Mon.Not.R.Astron.Soc.000,1–16(0000) Printed21January2011 (MNLATEXstylefilev2.2)
The pairwise velocity probability density function in
models with local primordial non-Gaussianity
1
Tsz Yan Lam1⋆, Takahiro Nishimichi1⋆ & Naoki Yoshida1⋆
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0 1 Institute for the Physics and Mathematics of the Universe, Universityof Tokyo, Kashiwa, Chiba 277-8583, Japan
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n
a 21January2011
J

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ABSTRACT

] We study how primordial non-Gaussianity affects the pairwise velocity probability
O
density function (PDF) using an analytical model and cosmological N-body simula-
C tions.Weadoptthelocaltypenon-Gaussianmodelscharacterizedbyfnl,andexamine
. boththelinearvelocitydifferencePDFandthelinearpairwisevelocityPDF.Weshow
h
p explicitlyhowfnl inducescorrelationsbetweenoriginallyindependentvelocitiesalong
- the parallel and the perpendicular to the line of separation directions. We compare
o
the model results with measurements from N-body simulations of the non-Gaussian
r
t models.LineartheoryfailstopredictthePDFinthefnl models.Thereforewedevelop
s
a an analytic model based on the Zeldovich approximation to describe the evolution of
[ the velocityPDF.Our analyticalmodel andsimulationresultsshowremarkablygood

agreementin both the parallel and the perpendicular directions for the PDF profiles,
2
v aswellasthechangeinthePDFduetoprimordialnon-Gaussianity.Theagreementis
6 particularlygoodfor relativelysmall separations(<10h−1Mpc). The inclusion of the
0 evolutionof the velocity PDF is important to obtaina good descriptionon the signa-
4
ture of primordial non-Gaussianity in the PDF. Our model provides the foundation
0
. to constrain fnl using the peculiar velocity in future surveys.
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0 Key words: methods:analytical-darkmatter-largescalestructureofthe universe
0
1
:
v
1 INTRODUCTION
i
X
Primordialnon-Gaussianityhasattractedmuchattentionowingtoitsabilitytodistinguishinflationarymodels(e.g.,Buchbinderet al.
r
a 2008;Khoury& Piazza2008;Silvestri & Trodden2008;Wands2010,andreferencestherein).Severalcosmologicalprobesusing
theCMB(Komatsu et al.2010;Hikage et al.2008;Yadav & Wandelt2008;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 & Komatsu2007;Izumi& Soda2007;Lo Verdeet al.2008;Dalal et 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 & Seljak2010a;Liguori et al.
2010;Tseliakhovich et al.2010;Xia et al.2010;Chongchitnan & Silk2010;Giannantonio & Porciani2010;Marian et al.2010;
Smith et al. 2010) havebeen considered.
Here,weintroduceanewlarge-scalestructureprobeofprimordialnon-Gaussianity–thepairwisevelocityPDF.Current
large-scalestructureprobesofprimordialnon-Gaussianityfocusontheclusteringofhalos,halo/voidabundances,bispectrum,
andthePDFofdarkmatterfield–allofthemarerelatedtothechangeofthedensityfieldduetoprimordialnon-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 with
the 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:[email protected]
2 T. Y. Lam, T. Nishimichi & N. Yoshida
distributionsoflinearvelocityfields.InparticularCatelan & Scherrer(1995)studiedthedistributionoftheparallelcomponent
ofthepairwisevelocityusinglineartheory.Lam et al.(2010)discussedtheeffectofprimordialnon-Gaussianityontheredshift
space distortion. They made use of the ellipsoidal collapse model to derive the modification in the Kaiser factor relating the
realandredshiftspacepowerspectra,withoutdiscussingthemodificationinthevelocityfield.Duringthepreparationofthis
work,Schmidt(2010)studiedtheprimordialnon-Gaussianitysignatureinthepeculiarvelocitiesofdensitypeaksusinglinear
theory. Our present study has two important improvements over these previous works: first we show that primordial non-
Gaussianityinducesacorrelationbetweeenvelocitiesintheparallelandtheperpendiculartothelineofseparationdirections.
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 f in the pairwise velocity
nl
PDF even at separation as big as 50 h−1Mpc. We use cosmological N-body simulations to show this. We thus develop an
analytic model to describe the evolution of the velocity PDF. Our model is based on the Zeldovich Approximation. We
illustrate theimprovement in theN-body measurement comparisons.
Thepeculiarvelocityfieldhasbeeninvestigatedasaprobeofcosmologyforbothdarkmatter(Gorski1988;Seto & Yokoyama
1998;Shethet al.2001;Kuwabara et al.2002;Scoccimarro2004)andbiasedtracers(Sheth& Diaferio2001;Sheth et al.2001;
Hamana et al. 2003; Sheth & Zehavi 2009). Other studies (Yoshidaet al. 2001; Peel 2006; Bhattacharya & Kosowsky 2007,
2008) discuss the possibility of constraining dark energy via the kinetic Sunyaev-Zeldovich effect. In this study we focus on
thepeculiar velocity field of the dark matter field. The effect of primordial non-Gaussianity on the velocity of biased tracers
will bediscussed in futurework.
We first describe how the linear velocity PDF changes due to primordial non-Gaussianity in section 2. Throughout the
present paper, we work with models with non-vanishing primordial bispectrum; we will use the local f type primordial
nl
non-Gaussianity toillustrate the calculations. The Bardeen potential Φ in the local f model is
nl
2 2
Φ=φ+f (φ −hφ i), (1)
nl
where φ is a Gaussian potential field and f is the nonlinear quadratic parameter. The above non-Gaussian correction is
nl
defined at z = z for this study. It has been suggested that mass weighting is important for the peculiar velocities
CMB
(Scoccimarro 2004; Sheth & Zehavi 2009). We will discuss the effect of primordial non-Gaussianity on both the uniform
weighted as well as themass weighted linear velocity PDF.In section 3we describe theanalytical model toapproximatethe
evolution of the pairwise velocity PDF. The theoretical predictions of both the linear theory and the analytical model are
compared with measurements from N-body simulations. Weconclude our findingsin section 4.
2 LINEAR VELOCITY PDF
2.1 Preliminary
The linear overdensity δ(k) and theBardeen potential Φ(k) is related bythe Poisson equation
2
δ(k,z)=D(z)k M(k)Φ(k), (2)
whereM(k)=2c2T(k)/3ΩmH02 andT(k)isthemattertransferfunction(notewedonotincludethek2 factorinM(k)).The
continuityrelates the linear overdensity and thepeculiar velocity u(k,z):
δ˙(k)+θ(k)=0, (3)
where θ(x)≡∇·u(x) is thedivergence of thevelocity field.The velocity field is described solely by its divegerence since its
vorticity decays dueto theexpansion of theuniverse (for example see Bernardeau et al. 2002). Hence
u (k)=iD˙(z)k M(k)Φ(k), (4)
j j
where i2 =−1 and the subscript j denotes the coordinate of the peculiar velocity. In this study we will be interested in the
relative velocities in the parallel (k) and perpendicular (⊥ and ⊥ ) to the line of separation for two particles separated by
a b
some distance r.
Connected moments higher than the second order vanish when theBardeen 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-vanishingbispectrum(includingthelocalf andtheequilateraltriangletypef ).Someotherstudiesinvestigatemodels
nl nl
with a non-vanishing trispectrum (the g model, see for example Desjacques & Seljak 2010b). While this work focuses on
nl
primordialnon-Gaussianitywithaleadinglynon-vanishingbispectrum,itcanbeextendedtostudymodelswheretheleading
order of the non-vanishingconnected moment is higher than the third order. In what follows we will use the local f model
nl
to demonstrate how the non-vanishing third order connected moment modifies the pairwise velocity PDF. The calculation
can also apply to models with other typesof primordial bispectrum.
We denotethe peculiar velocity at position x by u(x) and the relative velocity by v(r)≡u(x)−u(x′)=u−u′ where
Pairwise Velocity PDF in the f model 3
nl
x−x′ =r. We also denote thevelocity difference PDF p(v;r) as thePDF of thepeculiar velocity difference at two random
positionsseparatedbyr;thepairwisevelocityPDFq(v;r)asthePDFofthedifferenceofthepeculiarvelocitiesoftwotested
particlesseparatedbyr.ThelatterPDFisthepairweightedversionoftheformeroneandtheyareingeneralnotequivalent
(see, for example Scoccimarro 2004). For simplicity we sometimes call the former PDF the uniform weighted PDF and the
latter one themass/pair weighted PDF.
The numerical integration of the third order connected moments are carried out by Monte-Carlo integration using the
numerical package CUBA (Hahn 2005).
2.2 Linear Velocity Difference PDF
2.2.1 Case: f =0
nl
ThelinearvelocitydifferencePDFoftworandompositionsseparatedbyadistancerwhenf =0isgivenbythemultivariate
nl
normal distribution:
p0(v;r)= 1 exp −1vTA−1v , (5)
(2π)3/2 |A| 2
(cid:18) (cid:19)
where v = (vk,v⊥a,v⊥b) in which vk corresponds to the rpelative velocity parallel to the line of separation and v⊥a and v⊥b
arethetwovelocitydifferencesperpendiculartothelineofseparation.HereAisthecovariancematrix.Theaboveexpression
simplifiessincethereisnocorrelationbetweendifferentcomponentsofv.Hencether.h.sbecomesaproductofthreeunivariate
normal distributions:
p0(vk,v⊥a,v⊥b;r)=p0(vk;r)p0(v⊥a;r)p0(v⊥b;r), (6)
where thevariances of theunivariate normal distributions are hv2i and hv2i≡hv2 i=hv2 i respectively, and
k ⊥ ⊥a ⊥b
hvk2i= 3π12D˙02 dkPΦ(k)k4M2(k) 1−3j0(kr)+6j1k(krr) (7)
Z (cid:20) (cid:21)
hv2⊥i= 3π12D˙02 dkPΦ(k)k4M2(k) 1−3j1k(krr) . (8)
Z (cid:20) (cid:21)
Herej0 and j1 is thespherical bessel function, D˙0 =dD/dt is thetimederivativeof thelinear growth factor D at z =0and
PΦ(k)≈Pφ(k)istheBardeenpotentialpowerspectrum.Iostropymeansonecantransform therectangularcoordinateinthe
plane perpendicularto theline of separation into thepolar coordinate and write equation (6) as
p0(vk,v⊥a,v⊥b;r)dvkdv⊥adv⊥b =2πv⊥p0(vk;r)p0(v⊥;r)dvkdv⊥, (9)
wherev2 =v2 +v2 .Noticethatournotationshv ···i(=hv ···i=hv ···i)andv (= v2 +v2 )arenotequivalent.
⊥ ⊥a ⊥b ⊥ ⊥a ⊥b ⊥ ⊥a ⊥b
q
2.2.2 Case: f 6=0
nl
When f 6=0, the primordial bispectrum is non-zero and its functional form in thelocal typeis
nl
3
BΦ(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-vanishingconnected momentsare
2hvfk3i = (122πD˙)036 d3k1 d3k2P(k1)P(k2)M(k1)M(k2)M(k12)k1kk2kk12k[sin(k12kr)−2sin(k2kr)] (11)
nl Z Zcosµ2≥0
hv2kfv2⊥i ≡ hv2kfv⊥2ai = hv2kfv⊥2bi = (42Dπ˙)036 d3k1 d3k2P(k1)P(k2)M(k1)M(k2)M(k12)[k1⊥k2⊥k12ksin(k12kr)
nl nl nl Z Zcosµ2≥0
−2k1⊥k2kk12⊥sin(k2kr)+2k1⊥k2kk12⊥sin(k12kr)−2k1kk2⊥k12⊥sin(k2kr)−2k1⊥k2⊥k12ksin(k2kr)], (12)
wherek12 =|k1+k2|,kik =kicosµi,ki⊥ =kisinµicosφi,andk12k =k1k+k2k(similarlyfor⊥).Notethatconnectedmoments
involving odd power of v vanish irrespective of the value of f . The non-vanishing component hv v2i results in couplings
⊥ nl k ⊥
betweenvelocitiesintheparallelandtheperpendiculartothelineofseparationdirections.Asaresultthelinearmultivariate
PDFcannolongerbewrittenasaproductofthreeindependentunivariatePDFs.Weusethetri-variateEdgeworthexpansion
to approximate thelinear uniform weighted velocity difference PDF for f 6=0 (see appendix for derivation):
nl
p(vk,v⊥a,v⊥b;fnl,r)=p0(vk,v⊥a,v⊥b;r)[1+α300h300+α120(h120+h102)], (13)
where
α300 = 61hvh2vik33i/2, α120 = 21hv2hiv1k/v22⊥hvi2i,
k k ⊥
4 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 (thick
bluedottedcurve),−γ400 (bluepentagons),−γ220 (cyanhexagons),γ500 (thinmagentashort-long-dashedcurve),andγ320 thin(orange
dot-dashedcurve).fnl=−100forthoseparameters apply.α300,α120,−γ100,and−β100 areshiftedverticallyby-2forclarity.
and h ≡H (ν )H (ν )H (ν ) is theproduct of Hermite polynomials of different orders. In particular,
ijk i k j ⊥a k ⊥b
v3 v
h300 ≡νk3−3νk = hv2ik3/2 −3hv2ik1/2 (14)
k k
h120 ≡νk(ν⊥2a−1)= hv2vik1/2 hvv⊥22ai −1 (15)
k (cid:18) ⊥ (cid:19)
v v2
h102 ≡νk(ν⊥2b−1)= hvk2ik1/2 hv⊥2⊥bi −1!. (16)
Notethat Schmidt (2010) marginalized velocities perpendicular to theline of separation, essentially setting α120 =0.
Figure 1 shows thescale dependenceof 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 f becomes significant for big |ν | or |ν |.
nl k ⊥
Furthermoreα300 and α120 are in thesame order of magnitude, henceboth terms haveto beincluded in thecomputation of
thelinear peculiar velocity differencePDF.
Figure2showstheratiosofthelinearpeculiarvelocitydifferencePDFforf =100tothecorrespondingGaussianPDF
nl
at r =8 h−1Mpc. The axes labeled vll indicates the velocity in the parallel to the line of separation direction and vl is the
magnitudeofthevelocity perpendiculartothelineofseparation (recall vl = v2 +v2 ).Theupperpanelshowsboththe
⊥a ⊥b
contourand colormapsoverawiderangeofvelocity.Theeffectof primordialqnon-Gaussianity ismost significant at extreme
velocities. Both the contour and the color maps show variations in the ratio for different values of v at fixed v , indicating
⊥ k
thatthemodificationduetof isdegeneratedintheplaneof(vll,vl ).Whiletheupperpanelshowsthattheratiocanbeas
nl
bigas4,thefirstorderEdgeworthexpansionbreaksdowninsuchrarecases(about10-σ level)–itisevidentbythenegative
probability at the otherend of theplot.
Pairwise Velocity PDF in the f model 5
nl
4
3
4
2
3
2 1
1 0
0
-1
-1
-2 -2
-3 -3
2000
1500
-2000-1500 1000
-1000 -500 vl-
0 500 500
1000
vll 1500
2000 0
1.3 1.15
4.5 700 4.5 700
4 1.2 4 1.1
600 600
3.5 3.5
500 1.1 500 1.05
3 3
2.5 vl- 400 1 2.5 vl- 400 1
2 300 2 300
1.5 0.9 1.5 0.95
200 200
1 1
0.8 0.9
0.5 100 0.5 100
0 0 0.7 0 0 0.85
-1000 -500 0 500 1000 -1000 -500 0 500 1000
vll vll
-4 -2 0 2 4 -4 -2 0 2 4
Figure 2. Ratio of linear velocity difference PDF p/p0 inequation (13)for r=8h−1Mpc and fnl =100. vllindicates velocity inthe
paralleltothelineofseparationdirectionwhilevl isthemagnitudeofthevelocityperpendiculartothelineofseparation.Upperpanel
showstheratioforawiderangeofvelocity. Thelowerleftpanelzooms intoregionsforaround5-σ asindicatedbytheouteraxes. The
lowerrightpanelshowsthesameregionsasthelowerleftpanel,butexplicitlysettheparameterα120 =0aspreviousstudydid.
The lower panels of Figure 2 show the color maps of the same PDF ratio on a smaller velocity range to increase the
dynamic range of the effect of f on the linear velocity difference PDF. The left panel shows the modification obtained in
nl
thisstudy(thecorrectionterminequation(13))whiletherightpanelshowstheresultforsettinghv v2 i=0.Theouteraxes
k ⊥
ofthelowerpanelsshowtheσ-levelincorrespondingdirections.At5-σ level(vlldirection) thechangeisabout 30%,andthe
bigger the σ-level in the perpendicular direction the greater the modification is. The linear theory suggests that for f > 0
nl
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−1Mpc). While it is generally believed that linear theory applies
forlargescales, wewillshowinthenextsectionthatlineartheoryfailstodescribethechangeinthevelocityPDFduetof
nl
even at separation as big as 50 h−1Mpc.
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−1Mpc), the
linear theory velocity correlations of massive halos (in the parallel to the line of separation direction) is not consistent with
6 T. Y. Lam, T. Nishimichi & N. Yoshida
N-bodymeasurements(Croft & Efstathiou 1995).Sheth & Zehavi(2009)pointedoutthatthevelocitycorrelations ofbiased
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 studyfocuses on thedifference of the peculiar velocities of
unbiasedtracer,itisusefultoexaminewhetherthepairweightingwouldimprovethelineartheoryprediction.Itisreasonable
to include this mass weighting as the pairwise velocity PDF measured in N-body simulations is obtained by counting the
relativevelocitiesofsimulatedparticles.Asaresultregionswithmoreparticles(overdenseregions)haveabiggerweightthan
regions with less particles (underdenseregions).
2.3.1 Case: f =0
nl
Scoccimarro(2004)discussedhowtogeneralizethelinearvelocitydifferencePDFwhenf =0toincludethemassweighting.
nl
The resulting linear pairwise velocity PDFq0 is related to theassociated no weighting velocity difference PDFp0 by
[1+ξ(r)]q0(νk,ν⊥a,ν⊥b;r) =1+ξ(r)+h100β100+h200β200, (17)
p0(νk,ν⊥a,ν⊥b;r)
where δ≡δ(x),δ′ ≡δ(x+r), ξ≡ξ(r)=hδδ′i is thematter correlation function at separation r,and
hv δi hv δ′i hv δihv δ′i
β100 = hv2ki1/2 + hv2ki1/2, β200 = khv2ik , (18)
k k k
hvkδi=hvkδ′i= 2π12D˙0 dkPΦ(k)k4M2(k)[kj1(kr)]. (19)
Z
The r.h.s of equation (17) does not depend v⊥, hence q0 can still be written as a product of three univariate PDFs – the
correction termonther.h.sofequation(17)onlymodifiestheunivariatePDFofv .Thescale dependenceoftheparameters
k
−β100 (redsketealtriangularsymbols)andβ200 (thickcyanshort-long-dashedcurve)areshowninFigure1:theirmagnitudes
are big compared to others shown in the same figure since they are ’Gaussian parameters’ that do not dependon f .
nl
2.3.2 Case: f 6=0
nl
When f 6=0 additional terms contribute to the mass weighted linear velocity difference PDF. To the first order of f the
nl nl
expression is (see derivation in appendix)
q(ν ,ν ,ν ;f ,r)
[1+ξ(r)] k ⊥a ⊥b nl =1+ξ(r)+h100γ100+h200β200+h300γ300+(h120+h102)γ120
p0(νk,ν⊥a,ν⊥b;r)
+h400γ400+(h220+h202)γ220+h500γ500+(h320+h302)γ320, (20)
where
γ100 =β100+ hhvvk2δi1δ/′2i γ300 =α300(1+ξ(r))+ β1200 hvhvk22δi′i γ120 =α120(1+ξ(r))+ β1200 hhvv2⊥2δi′i
k k ⊥
β100
γ400 =α300 γ220 =α120β100 γ500 =α300β200
2
4 2 5 3
γ320 =α120β200 h400 =νk−6νk+3 h500 =νk −10νk +15νk,
and for thelocal f typeprimordial non-Gaussianity
nl
hv2kfδδ′i =−(42Dπ˙)06 d3k1 d3k2P(k1)P(k2)M(k1)M(k2)M(k12)[k1kk22k122sin(k12kr)+k12k22k12ksin(k2kr)
nl Z Zcosµ2≥0
2 2
−k1kk2k12sin(k2kr)] (21)
h2vfk2δ′i = (22Dπ˙)026 d3k1 d3k2P(k1)P(k2)M(k1)M(k2)M(k12)[2k1kk22k12kcos(k2kr)−k1kk2kk122cos(k12kr)
nl Z Zcosµ2≥0
2 2 2 2 2
+2k1kk2kk12cos(k2kr)−2k1kk2k12kcos(k1kr)−2k1kk2k12kcos(k12kr)+2k1kk2k12k−k1kk2kk12] (22)
hv22⊥fδ′i = (22Dπ˙)026 d3k1 d3k2P(k1)P(k2)M(k1)M(k2)M(k12)[2k1⊥k22k12⊥cos(k2kr)−k1⊥k2⊥k122cos(k12kr)
nl Z Zcosµ2≥0
2 2 2 2 2
+2k1⊥k2⊥k12cos(k2kr)−2k1⊥k2k12⊥cos(k1kr)−2k1⊥k2k12⊥cos(k12kr)+2k1kk2k12k−k1kk2kk12]. (23)
Thescaledependencesofγ areshowninfigure1.AsinthecaseofuniformweightedlinearPDF,thelinearpairwisevelocity
ijk
PDFcan no longer be written as a product of threeindependentPDFs when f 6=0 dueto thenon-vanishingterms hv v2i
nl k ⊥
andhv2δ′i.Inaddition,whenonesetsδ=δ′ =0,theaboveexpressionrecoverstheuniform weightedlinearPDFforf 6=0
⊥ nl
(equation (13)).
Figure3showsthemodificationfactorofthelinearpairwisevelocityPDFduetofnl (q/q0 inequations(17)and(20)).As
Pairwise Velocity PDF in the f model 7
nl
4
3
4
2
3
2 1
1 0
0
-1
-1
-2 -2
-3 -3
2000
1500
-2000-1500 1000
-1000 -500 vl-
0 500 500
1000
vll 1500
2000 0
1.3 1.2
4.5 700 1.2 4.5 700 1.15
4 600 1.1 4 600 1.1
3.5 3.5 1.05
500 1 500
3 3 1
0.9
2.5 vl- 400 2.5 vl- 400 0.95
0.8
2 300 2 300 0.9
1.5 0.7 1.5 0.85
200 200
1 0.6 1 0.8
0.5 100 0.5 0.5 100 0.75
0 0 0.4 0 0 0.7
-1000 -500 0 500 1000 -1000 -500 0 500 1000
vll vll
-4 -2 0 2 4 -4 -2 0 2 4
Figure3.Similarplottofigure2butfortheratioofthelinearpairwisevelocityPDF(q/q0inequations(17)and(20)).Thelowerright
panelsetsα120=γ220=γ320=hv2 δ′i=0.
⊥
inFigure2theupperpanelshowstheratio inawiderangeofvelocity,includingregionswith negativeprobabilityindicating
thebreakdownof thefirstorderEdgeworth expansionapproximation. Thelowerleft panelenlarges thecolor mapon regions
about5-σ levelandtheouteraxeslabeltheσ-level.Thelowerrightpanelshowstheratioifoneexplicitlysetsallthirdorder
moments involving v⊥ to zero (i.e. α120 =γ220 = γ320 = hv2⊥δ′i =0). The effect of fnl in the mass weighted linear pairwise
PDFhas thesame trend as theuniform weighted PDF: theeffect of f is degenerated in theparallel and theperpendicular
nl
tothelineof separation directions; and thegeneral trendobserved in uniform weighted linear PDFstill applies – forf >0
nl
theprobabilityoffindinginfalling pairsincreaseswhiletheprobabilityoffindingpairsmovingapart decreases.Theinclusion
of the mass weighted quantites γ strengthens the effect of f and it is most significant in the decrement in probability of
ijk nl
outgoing pairs.
3 EVOLVED VELOCITY PDF
GiventhelinearvelocityPDFwenowdescribeamodelfortheevolutionofthePDF.WeadopttheZeldovichApproximation
which assumes the comoving velocity remains unchanged. At some initial redshift z , if two particles separated by r have a
i i
relative velocity (vki,v⊥ia,v⊥ib). Then at a later redshift z = z0 (≪ zi) the distances traveled are D0/D˙i(vki,v⊥ia,v⊥ib) (note
8 T. Y. Lam, T. Nishimichi & N. Yoshida
that it assumes D0 ≫Di).Hence theseparation of the two particles becomes
2 2
r2 = r + D0vi + D0 (vi 2+vi 2). (24)
i D˙ k D˙ ⊥a ⊥b
(cid:18) i (cid:19) (cid:18) i(cid:19)
The evolved relative velocities (with respect to theupdated position) are:
v = D˙0 rivki + D0vi2 (25)
k r D˙i D˙i2 !
|v |2 =v2 +v2 = D˙0vi 2−v2, (26)
⊥ ⊥a ⊥b D˙ k
(cid:18) i (cid:19)
where vi2 =vi2+vi 2+vi 2.The evolved velocity difference PDF is therefore
k ⊥a ⊥b
r2
p(Vk,V⊥;R)= dridvkidv⊥iadv⊥ibRi2p(vki,v⊥ia,v⊥ib;r)δD(r−R)δD(vk−Vk)δD(v⊥−V⊥), (27)
Z
(see,forexample,Seto & Yokoyama1998)wherethedirac-deltafunctionsuseequations(24),(25),and(26)tomaptheinitial
quantities to the evolved one. In the following we apply the above formula to compute the evolved velocity difference PDF
from equations(6)and(13)andcomparetheresultstomeasurementsfrom N-bodysimulations. Wesetzi andz0 tobe1100
and 0.5 respectively, justifying theapproximation D0 ≫Di.
3.1 Comparisons to N-body measurements
WemeasuredthepairwisevelocityPDFfromasetofN-bodysimulationsdescribedinNishimichi et al.(2009).Thesimulation
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
simulationwasperformedinaboxof2000h−1Mpconaside,containing5123 particleswhosemassis4.6×1012 h−1M .The
⊙
measurementsweremadefromthesimulationoutputatz=0.5.ToaccountforthediscretenatureofN-bodymeasurements,
separation r in the following refers to two particles having a separation between (r−2,r+2) in the unit of h−1Mpc. For
thetheoreticalmodelweapplythesameseparation selection inthedirac-deltafunctionofr.Thecomparisonsofthevelocity
PDFofv and v arepresented inthenexttwosubsections. Wewill thendiscusstheimplications of thecomparisons inthe
k ⊥
next section.
3.1.1 Parallel to the line of separation: p(v ;r,f )
k nl
Figure4showsthePDFcomparisonsofthevariousanalyticalpredictionstotheN-bodymeasurementsfortherelativevelocity
parallel to the line of separation for 4 different separations (4, 8, 12, 50 h−1Mpc). The upper panel in each subfigure shows
thePDFprofileforf =0:solidsymbolsaretheN-bodymeasurements,cyandottedcurvesaretheanalyticalpredictionsof
nl
equation (27) (marginalized over v ), green dot-long dashed (mass weighted) and magenta long dashed (uniform weighted)
⊥
curvesare the linear theory predictions respectively.
The predictions from our analytical evolution model give good matches to the N-body measurements at all scales we
investigated, except at large outgoing velocities (i.e. v ≫0). The agreements with the N-body measurements are better in
k
small separations (4 and 8 h−1Mpc) than large separation (12 h−1Mpc). The analytical predictions of ourmodel are able to
describe the high infalling velocity regimes (v < −1000). This is very encouraging since one may expect that high velocity
k
regionsarenot describedbyoursimplemodel.Theanalytical modelalsopredictsknee-shapetransitionsat around300, 600,
and 900km/s for r = 4, 8, 12 h−1Mpc respectively. Similar but less significant knee-shape changes can be seen from the
N-bodymeasurementsaroundthecorrespondingvelocities. Inthehighoutgoingvelocityregionsthematchisnotasgood as
theinfalling regimes. Webelievethedisagreement in theoutgoing velocity regime as compared totheinfalling regime is due
to non-linear evolution of velocity that is not described by our current model – originally infalling pairs eventually become
outgoing(seeequation(25))andnon-linearinteractionsbetweenparticlesinclosecontact arenotdescribedbytheZeldovich
Approximation.
On the other hand, the linear theory predictions do not fare as well as the analytical model’s. At small separations the
predictionsofthelinearvelocitydifferencePDFdonotmatchtheN-bodymeasurements–themagnetacurvesmissboththe
extrema and thepeaks of thePDF at separation r= 4, 8, and 12 h−1Mpc. This is consistent with previous studies that the
lineartheoryuniformweightedvelocityPDFdoesnotagreewiththevelocitycorrelationintheparalleltothelineofseparation
direction. On the contrary the mass weighted linear predictions provide reasonable matches to the N-body measurements
nearthepeaksof thePDFat different separations. Itconfirmsthefindingof Sheth & Zehavi(2009): thevelocity correlation
can bedescribed bythelinear theory whenthemass weighting istaken intoaccount.Thematchingof thepeaksofthePDF
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 |v | is of the order of
k
Pairwise Velocity PDF in the f model 9
nl
(a)r=4Mpch−1 (b)r=8Mpch−1
(c)r=12Mpch−1. (d)r=50Mpch−1
Figure 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 the
evolutionmodelbasedontheZeldovichapproximation(equation(27)),greendot-longdashedcurvesarethelinearpairwisevelocity(q0
inequation (17)), and magenta long dashed curves arethe linear velocity difference PDF(equation (6)). Lower panels show the ratios
of 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)andbluedot-shortdashedcurves(fnl=−100)arepredictionsofequation(27),greendot-longdashedcurvesaretheratios
forthelinearpairwisePDF(fnl=100),andmagentalongdashedcurvesaretheratiosforthelinearvelocitydifferencePDF(fnl =100).
Redshort-long-dashed curves show the prediction fromequation (27), neglecting thirdorder moments involving velocity perpendicular
tolineofseparation(seethelowerrightpaneloffigure2).
a few hundreds km/s. The disagreement is getting worse as the separation decreases. At large separation (50 h−1Mpc), the
two linear theory predictions are very similar and they describe the N-body measurements well. They are very closed to the
prediction of theanalytical model and thematch to themeasurements 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 f =±100 to the
nl
associated PDFs for f =0 at different separations. Measurements from the N-body simulations are represented by crosses
nl
(f =100)andsquares(f =−100).OthercurvesinthelowerpanelsshowthePDFratiosofdifferentanalyticalpredictions:
nl nl
10 T. Y. Lam, T. Nishimichi & N. Yoshida
cyan dotted (f =100) and blue dot-short dashed (f =−100) curves are the predictions of the evolution model based on
nl nl
the Zeldovich Approximation (equation (27)); red short-long-dashed curves show the similar predictions for f = 100 but
nl
setting hv v2 i = 0 (see the lower right panel of figure 2 for the corresponding linear theory comparison at r = 8 h−1Mpc).
k ⊥
The other two curves are the linear theory predicted ratios for f = 100: the magneta long dashed curves are the uniform
nl
weighted velocity predictions and thegreen dot-long dashed curvesare themass weighted velocity predicted ratios.
For the range of separations we investigated the change in the pairwise velocity PDF due to f = ±100 is at most 5%
nl
in the infalling velocity regime and 10% in the outgoing velocity regime. In contrary to the linear theory predictions (see
figures2and3,andthemagentalongdashed aswell asthegreendot-longdashedcurves),theeffectofpositivef enhances
nl
the probability of having pairs in both the infalling and the outgoing high velocity ends. This is true for all the separations
welook at, from 4to50 h−1Mpc– scales in which thelinear theory is believed tobevalid. Wealso checkusinga simulation
with a smaller box (1 h−1Gpc on a side) to estimate the box size effect: shrinking the box volume to one-eighth of its size
changes thePDF ratios byat most 2% at v =2000 km/s.
k
Thepredictionsofourevolutionmodel,regardlessofwhetheroneexplicitlysethv v2 i=0,matchtheN-bodymeasure-
k ⊥
ments reasonably well when r = 4 h−1Mpc. 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 f = 100 it gradually
nl
decreases from positive to negative, then changes direction and become positive again. The same transition is also observed
at larger separations and the model’s predictions are able to match the N-body measurements. In contrast the ratios of the
lineartheorypredictions(boththeuniformandthemassweighted)failtomatchtheN-bodymeasurementsinbothinfalling
andoutgoingvelocityregimes.WhilethelineartheorypredictedratiosstillhavethesametrendastheN-bodymeasurements
in the infalling velocity regime (but the differences between the predictions and the measurements are large), the predicted
ratiosfromlineartheoryareoppositetothemeasurementsintheoutgoingvelocityregime(thesameisalsoobservedatlarger
separations). Hencethe linear theory cannot be used to predict the effect of f on thepairwise velocity PDF.
nl
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. Nonetheless
the predictions still qualitatively match the measurements at r =8 and 12 h−1Mpc. At r =50 h−1Mpc, on the other hand,
the analytical predictions converge to the linear theory predictions and show opposite trends to the measurements. In the
outgoing velocity region, there are some interesting transitions in the analytical model predictions: at r = 12 h−1Mpc and
f = 100, the predicted ratios (the cyan as well as red curves) first show a decrement at small v . They then turn around
nl k
near v =800km/s and keep increasing for larger v . Such transitions are not seen in theN-body measurements.
k k
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 the
measurements at large separations in the nextsection.
3.1.2 Perpendicular to the line of separation: p(v ;r,f )
⊥ nl
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 f = 0 and the lower panels show the ratios of the PDF for
nl
f = ±100 to the associated PDF for f = 0. The symbol and curve labelings are the same as in figure 4, but we do not
nl nl
showthePDFratiopredictions ofthelinear theoryin thelower panels sincethelinear theory predictsnoeffect of f inthe
nl
marginalized PDF, that is p(v ;r,f )/p(v ;r,f =0)=1.
⊥ nl ⊥ nl
ThepredictionsofthePDFprofilefromouranalyticalmodel(equation(27))matchtheN-bodymeasurementsreasonably
well, except around v = 0 at r = 50 h−1Mpc. 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
⊥ ⊥
lineartheorypredictions,ontheotherhand,failtomatchthemeasuredPDFforr=4,8,and12h−1Mpc.Atr=50h−1Mpc
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. Notethat there is no difference between the mass weighted and theuniform weighted linear theory predictions
⊥
since thecorrection term dueto mass weighting does not dependon 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 f = 100 (−100). It then gradually
⊥ ⊥ nl
switches direction and at high v show an increment (decrement). This gradual switch from decrement to increment is
⊥
most significant at small separation. Our analytical model is able to predict the change in the PDF due to primordial non-
Gaussianity qualitatively. Its prediction matches the gradual change of the PDF ratios (from decrement to enhancement for
f = 100, opposite for f = −100) and provides a rough approximation on the velocity where the crossing across unity
nl nl
occurs.HoweverthepredictionsalwaysunderestimatethePDFratiosinlowv regime.Athigherv themodelpredictionon
⊥ ⊥