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arXiv:1001.1154v1 [astro-ph.CO] 7 Jan 2010

Statistical Determination of Bulk Flow Motions

1Yong-Seon Song,1,2Cristiano G. Sabiu,1Robert C. Nichol and2Christopher J. Miller∗

1Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK

2Department of Physics & Astronomy, University College London, Gower Street, London, U.K

3Cerro-Tololo Inter-American Observatory, National Optical Astronomy Observatory,

950 North Cherry Ave., Tucson, AZ 85719, USA

(Dated: January 7, 2010)

We present here a new parameterization for the bulk motions of galaxies and clusters (in the linear

regime) that can be measured statistically from the shape and amplitude of the two–dimensional

two–point correlation function. We further propose the one–dimensional velocity dispersion (vp) of

the bulk flow as a complementary measure of redshift–space distortions, which is model–independent

and not dependent on the normalisation method. As a demonstration, we have applied our new

methodology to the C4 cluster catalogue constructed from Data Release Three (DR3) of the Sloan

Digital Sky Survey. We find vp = 270+433km/s (also consistent with vp = 0) for this cluster sample

(at ¯ z = 0.1), which is in agreement with that predicted for a WMAP5–normalised ΛCDM model

(i.e., vp(ΛCDM) = 203km/s). This measurement does not lend support to recent claims of excessive

bulk motions (≃ 1000 km/s) which appear in conflict with ΛCDM, although our large statistical

error cannot rule them out. From the measured coherent evolution of vp, we develop a technique to

re-construct the perturbed potential, as well as estimating the unbiased matter density fluctuations

and scale–independent bias.

PACS numbers: draft

I.INTRODUCTION

A decade ago, astronomers discovered the expansion

of the Universe was accelerating via the cosmological

dimming of distant supernovae [1, 2]. Since then, the

combination of numerous, and diverse, experiments has

helped to establish the Cosmological Constant (specifi-

cally a ΛCDM model) as the leading candidate to explain

this cosmic acceleration. However, with no theoretical

motivation to explain the required low energy vacuum of

the ΛCDM model, there is no reason to preclude alter-

native models, especially those based upon the possible

violation of fundamental physics which have yet to be

proven on cosmological scales [3, 4].

In addition to using geometrical probes like Super-

novae to constrain the cosmic acceleration, tests based

on the formation of structures in the Universe also pro-

vide a method for validating our cosmological models.

In particular, we can investigate the consistency between

the geometrical expansion history of the Universe and

the evolution of local density inhomogeneities to help re-

veal a deeper understanding of the nature of the cosmic

acceleration [5–9].

In general, there are three observables that can be used

to quantify structure formation in the Universe, namely

geometrical perturbations, energy–momentum fluctua-

tions and peculiar velocities, all of which will be measured

to high precision via future experiments like DES, LSST,

JDEM and Euclid (see details of these experiments in the

recent FoMSWG report [10]). In more detail, such weak

lensing experiments measure the integrated geometrical

∗Electronic address: yong-seon.song@port.ac.uk

effect on light as its trajectory is bent by the gravitational

potential. Likewise, galaxies (and clusters of galaxies)

measure the correlations amongst large–scale local inho-

mogeneities, while the observed distortions in these cor-

relations (in redshift–space) can be used to extract infor-

mation about peculiar velocities [11–14]. In this paper,

we explore the cosmological constraints on the physics of

cosmic acceleration using peculiar velocities, as it is one

of the key quantities required for a consistency test of

General Relativity [15, 16].

Early observational studies of the peculiar velocity

field, or “bulk flows”, have produced for many years dis-

crepant results [17], primarily due to small sample sizes

and the heterogeneous selection of galaxies. However, a

recent re-analysis of these earlier surveys [18] has now

provided a consistent observational picture from these

data and finds significant evidence for a larger than ex-

pected bulk motion. This is consistent with new mea-

surements of the bulk motion of clusters of galaxies us-

ing a completely different methodology[19, 20], which

leads to the intriguing situation that all these measure-

ments appear to be significantly greater in amplitude,

and scale, than expected in a concordance, WMAP5–

normalised ΛCDM cosmological model. Such discrepan-

cies with ΛCDM may give support to exotic cosmological

models like modified gravity [21].

Given the importance of these large–scale bulk flow

measurements, we propose here an alternative methodol-

ogy to help check these recent claims of anomalously high

peculiar velocities which are inconsistent with the stan-

dard ΛCDM cosmology. We start by outlining a statis-

tical determination of bulk flow motions using redshift–

space distortions in large-scale galaxy or cluster surveys.

Such redshift-space distortions are easily seen in the two–

dimensional correlation function (ξs(σ,π)), which is the

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decomposition of the correlation function into two vec-

tors; one parallel (π) to the line–of–sight and the other

perpendicular (σ) to the line–of–sight. On small scales,

any incoherent velocities of galaxies within a single dark

matter halo (or cluster) will just add to the cosmological

Hubble flow thus causing the famous “Fingers-of-God”

(FoG) effect which stretches the 2-D correlation function

preferentially in line-of-sight (π) direction. These dis-

tortions depend on the inner dynamics and structure of

halos and therefore, any cosmological information is dif-

ficult to distinguish from the halo properties. However,

on large scales (outside individual dark matter halos), the

peculiar velocities become coherent and follow the linear

motion of the matter thus providing crucial information

on the formation of large-scale structure [22].

In this paper, we compare predictions for ξs(π,σ)

to observations based on the C4 cluster catalogue [23]

from the Sloan Digital Sky Survey (SDSS) [24].

ing the Kaiser formulation [22], a theoretical model for

ξs(π,σ) is fit to the measured 2-D correlation function

in configuration–space, with the ξs parameterised by a

shape dependent part and a coherent evolution compo-

nent. We also propose that the 1-D linear velocity disper-

sion (vp) is a interesting quantity to report when measur-

ing redshift–space distortions, and complementary to tra-

ditional quantities like β, f or fσ8discussed recently [11–

13], as it is independent of both bias and the normalisa-

tion method. Therefore, the measured vp provides an

unbiased tracer of the evolution of structure formation.

Us-

II.STATISTICAL DETERMINATION OF

PECULIAR VELOCITY

The redshift–space two-point correlation function of

mass tracers (ξs(σ,π)) is an anisotropic function [22].

On small scales, it is elongated in the π-direction by the

“Fingers-of-God” effect, while on large scales, the gravi-

tational infall into overdense regions preferentially com-

presses the correlation function in the σ direction. There-

fore, peculiar velocities can be statistically measured by

analyzing the observed anisotropic pattern of ξs(σ,π) in

both the linear and non-linear regimes.

ξs(σ,π) is derived from the convolution of ξ(r) with

a probability distribution function of peculiar velocities

along the line of sight, which is usually called the stream-

ing model [25]. Even with the simplest form of a Gaussian

probability distribution, the streaming model describes

the suppression effect on ξs(σ,π) on small scales.

In the linear regime, the density fluctuations and pe-

culiar velocity are coherently evolved through the conti-

nuity equation, which is known as the Kaiser limit. Thus

the known correlation function of ξ(r) from the linear

perturbation theory developed by gravitational instabil-

ity is uniquely transformed into ξs(σ,π) [26–30].

The large scale limit of the streaming model is con-

sistent with the Kaiser limit [22], when both the density

and peculiar velocity fields are treated as statistical quan-

FIG. 1: ∆ξ∗

panel, we show the change in ξ∗

lower panel, we show the change in ξ∗

thick black curves are based on WMAP priors, while the thin blue

curves are for Planck prior. On the y-axis, we focus on the range of

scales probed by recent and planned reshift–space distortion mea-

surements.

l(r) for various CMB experiments priors. In the upper

l(r) for variations in ωm. In the

l(r) as a function of nS. The

tities [31]. This consistency test was developed further

to show that, even in the Kaiser limit, the description

of ξs(σ,π) in linear theory can be modified due to the

correlation between the ”squashing” (in the σ direction)

and dispersion effects (in the π direction) [32].

the assumption of a Gaussian pair-wise velocity distribu-

tion function, the dispersion effect smears into the Kaiser

limit description of ξs(σ,π) at around the percent level

which for our present work can be ignored. Thus we

adopt the Kaiser limit for the description of ξs(σ,π) in

linear regime while considering dispersion effect as a sys-

tematic uncertainty. We introduce below a new param-

eterisation of ξs(σ,π) in terms of the cosmological pa-

rameters and construct a method to measure the mean

velocity dispersion vpin a model independent way.

With

A.Model independent parameterisation of power

spectra

The discovery of cosmic acceleration has prompted

rapid progress in theoretical cosmological research and

prompted many authors to propose modification to the

law of gravity beyond our solar system. For example,

some theories based upon General Relativity can be mod-

ified by screening (or anti-screening) the mass of gravi-

tationally bound objects [3], while others include a non-

trivial dark energy component (e.g. interacting dark en-

ergy [33, 34], or clumping dark energy [35]) thus break-

ing the dynamical relations between density fluctuations

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and peculiar velocity in the simplest dark energy mod-

els. These theoretical ideas motivate us to express various

power spectra of the density field in a more convenient

way to test such theories.

We assume a standard cosmology model for epochs ear-

lier than the last scattering surface, and that the coherent

evolution of structure formation from the last scattering

surface to the present day is undetermined due to new

physics relevant to the cosmic acceleration. Thus we di-

vide the history of structure formation into two regimes;

epochs before matter-radiation equality (aeq) and a later

epoch of coherent evolution of unknown effect on struc-

ture formation from new physics.We can then express

various power spectra of the density field splits into these

two epochs, with the shape-dependent part determined

by knowledge of our standard cosmology, and the coher-

ent evolution part only affected by new physics. Mathe-

matically, this is written as,

PΦΦ(k,a) = DΦ(k)g2

Pbb(k,a) = Dm(k)g2

PΘmΘm(k,a) = Dm(k)g2

Φ(a),

b(a),

Θm(a), (1)

where Φ denotes the curvature perturbation in the New-

tonian gauge,

ds2= −(1 + 2Ψ)dt2+ a2(1 + 2Φ)dx2, (2)

and Θm denotes the map of the re-scaled divergence of

peculiar velocity θm as Θm = θm/aH.

spectra are then partitioned into a scale–dependent part

(DΦ(k),Dm(k)) and a scale-independent (coherent evolu-

tion) component (gΦ, gb, gΘm). We define here gb= bgδm

where b is the standard linear bias parameter between

galaxy (or cluster) tracers and the underlying dark mat-

ter density.

The shape of the power spectra is determined be-

fore the epoch of matter–radiation equality. Under the

paradigm of inflationary theory, initial fluctuations are

stretched outside the horizon at different epochs which

generates the tilt in the power spectrum. The predicted

initial tilting is parameterised as a spectral index (nS)

which is just the shape dependence due to the initial

condition. When the initial fluctuations reach the coher-

ent evolution epoch after matter-radiation equality, they

experience a scale-dependent shift from the moment they

re-enter the horizon to the equality epoch. Gravitational

instability is governed by the interplay between radiative

pressure resistance and gravitational infall. The different

duration of modes during this period results in a sec-

ondary shape dependence on the power spectrum. This

shape dependence is determined by the ratio between

matter and radiation energy densities and sets the loca-

tion of the matter-radiation equality in the time coordi-

nate. As the radiation energy density is precisely mea-

sured by the CMB blackbody spectrum, these secondary

shape dependences are parameterised by the matter en-

ergy density ωm= Ωmh2. Both of these parameters are

now well–determined by CMB experiments.

These power

The shape factor of the perturbed metric power spectra

DΦ(k) is defined as

DΦ(k) =2π2

k3

9

25∆2

ζ0(k)T2

Φ(k)(3)

which is a dimensionless metric power spectra at aeq,

where ∆2

ζ0(k) is the initial fluctuations in the comov-

ing gauge and TΦ(k) is transfer function normalized at

TΦ(k → 0) = 1. The primordial shape ∆2

on nS, as ∆2

plitude of the initial comoving fluctuations at the pivot

scale, kp= 0.002Mpc−1. The intermediate shape factor

TΦ(k) depends on ωm. The shape factor for matter fluc-

tuations and peculiar velocities Dm(k) are given by the

conversion from DΦ(k) of,

ζ0(k) depends

Sis the am-

ζ0(k) = A2

S(k/kp)nS−1, where A2

Dm(k) ≡4

9

k4

∗ω2

H4

m

DΦ(k),(4)

where H∗= 1/2997Mpc−1.

Unlike the shape part, the coherent evolution compo-

nent, gΦ, gb and gΘmare not generally parameterized

by known standard cosmological parameters. We thus

normalize these growth factors at aeqsuch that,

gΦ(aeq) = 1,

gδm(aeq) = aeqgΦ(aeq),

gΘm(aeq) = −dgδm(aeq)

dlna

. (5)

Instead of determining growth factors using cosmologi-

cal parameters, we measure these directly in a model-

independent way at the given redshift without referenc-

ing to any specific cosmic acceleration model and with

the minimal assumption of coherent evolution of modes

after aeq. Considering the uncertainty in the determi-

nation of A2

Sfrom the CMB anisotropy, which is degen-

erate with the optical depth of re-ionization, we com-

bine both A2

Sand gX (where X denotes each compo-

nent of Φ, b and Θm) with proper scaling for conve-

nience as g∗

S. Throughout this paper, we

use A∗ 2

WMAP5 results. Our result on measuring the bulk flow

motion is independent of our choice of an arbitrary con-

stant A∗ 2

S.

X= gXAS/A∗

S = 2.41 × 10−9for the mean A2

Svalue from the

B.Correlation function in the configuration space

In the linear regime of the standard gravitational in-

stability theory, the Kaiser effect (the observed squeezing

of ξs(σ,π) due to coherent infall around large–scale struc-

tures) can be written in configuration space as,

ξs(σ,π)(a) =

?

?4

8

35g∗2

g∗2

b

+1

3g∗

bg∗

Θm+1

5g∗2

?

Θm

?

ξ∗

0(r)P0(µ)

−

3g∗

bg∗

Θm+4

7g∗2

Θm

ξ∗

2(r)P2(µ)

+

Θmξ∗

4(r)P4(µ), (6)

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FIG. 2: The ξs(σ, π) correlation function. We plot for three con-

tours of ξs(σ, π) = 1,0.1,0.01 (from the inner to outer contour).

In the upper panel, the solid curves are for a ΛCDM cosmology,

while the dash and dotted curves are for models with gb= 1.5 and

2 respectively. In the lower panel, the solid curves are for a ΛCDM

cosmology, while the dash and dotted curves are for models with

gΘm= 1.5 and 2 respectively.

where Pl(µ) is the Legendre polynomial and the spherical

harmonic moment ξ∗

l(r) is given by,

ξ∗

l(r) =

?

k2dk

2π2D∗

m(k)jl(kr),(7)

where jlis a spherical Bessel function and ∗ denotes scal-

ing of the shape factor with A∗ 2

As discussed above, we ignore the effect on ξs(σ,π) of

the small-scale velocity dispersions within a single dark

matter halo [32] as the effect is only a few percent, and

split Eqn. 6 into a shape–dependent part (ξ∗

is determined by the cosmological parameters (nS and

ωm), and a coherent evolution component, which is pa-

rameterised by g∗

Θmiat the targeted redshift zi. The

shape part is therefore almost completely determined by

CMB priors, while the coherent evolution of structure

formation can be determined from fitting ξs(σ,π), in

redshift–space, as a function of redshift.

In Figure 1, we present the effect of CMB priors on

the value of ξ∗

l(r). In the top panel of Fig. 1, we provide

the expected variation in ξ∗

l(r) from varying ωm. We see

that varying aeq causes greater tilting in the shape of

ξ∗

l(r), since larger scale modes can come into the hori-

zon earlier. In addition to this contribution, the overall

amplitude of ξ∗

l(r) depends on ωmby a weighted trans-

formation between DΦ(k) and Dm(k). Considering the

marginalisation over CMB priors, we expect a discrep-

ancy of ≃ 5% with WMAP5 measurements, and just a

S.

l(r)), which

bi, g∗

few percent effect with the projected Planck priors.

In the bottom panel of Fig. 1, the dependence of ξ∗

on nSis given for both WMAP5 and Planck priors. The

overall shifting can be re-scaled by adjusting the pivot

point to the effective median scale of the survey. With

the measured WMAP5 prior of ∆nS= 0.015 [36], we ex-

pect variations of a few percent on the shape, while for an

estimated Planck prior of ∆nS= 0.0071 [37], we expect

ξ∗

affected maximally during the intermediate epoch, from

horizon crossing to the matter-radiation epoch. The de-

cay rate of the inhomogeneities differs by the ratio be-

tween matter and radiation energy densities.

Once CMB constraints are placed on the shape part

of Eqn. 6, the coherent history of structure formation is

obtained from the anisotropic moment of ξs(σ,π). Even

though both g∗

Θmiweight the evolution sector

simultaneously, their contribution to ξs(σ,π) are differ-

ent, which enables us to discriminate g∗

the monopole moment, g∗

biis the dominant component

since g∗

Θmiunless their is an excessive bulk flow.

Thus the variation of g∗

bigenerates a near isotropic am-

plification as illustrated in the top panel of Figure 2. In

the quadrupole moment, the cross-correlation between

δm and Θm is leading order. The reversed sign of the

quadrupole moment results in the squashing effect, and

it is sensitive to the variation of g∗

correlation is the leading order.

of Figure 2 the variation of g∗

the anisotropic moment. It is this signal which allows

both g∗

Θmito be probed separately using the

anisotropic structure of ξs(σ,π). The contribution from

the term having peculiar velocity autocorrelation is not

significant if excessive bulk flows are excluded.

l(r)

l(r) to be nearly invariant to nS. The shape of ξ∗

l(r) is

b iand g∗

Θmifrom g∗

b i. In

bi> g∗

Θmias the cross-

In the bottom panel

Θmimainly contributes to

biand g∗

C.Implication for cosmology from measuring g∗

and g∗

Θmi

bi

A measurement of g∗

fσmass

8

[14] and therefore, an excellent test of dark en-

ergy models (where f is the logarithmic derivative of

the linear growth rate and σmass

8

mass fluctuation in spheres with radius 8h−1Mpc). While

cosmological test of g∗

Θmiare free from bias, which is

notoriously difficult to measure accurately in a model–

independent way, the reported value of g∗

on the normalization which is also poorly constrained

(i.e., primordial amplitude) or model–dependent (i.e.,

σmass

8

).

Thus, we introduce a more convenient parameterisa-

tion of peculiar velocity which is independent of these

normalization issues. The measured g∗

bin i) which can be translated into the one-dimensional

(1-D) velocity dispersion in that redshift bin (vi

Θmiis equivalent to the quantity

is the root-mean-square

Θmidoes depend

Θmi(in the redshift

p) by,

vi2

p = g∗2

Θmi

H2

3

?∞

0

dk

kDm(k)dk.(8)

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5

σ (Mpc h−1)

π (Mpc h−1)

−60−40−200 204060

−60

−40

−20

0

20

40

60

−2−1.5−1−0.50 0.5

FIG. 3: The 2-D two–point correlation function (ξs(σ, π)) for the

SDSS DR3 C4 cluster survey with a median redshift of ¯ z=0.1. The

contours have been slightly smoothed.

In this formula, there is a degeneracy between g∗

g∗

Θmiwhich cannot be solely broken by fitting ξs(σ,π);

instead we simultaneously fit for vi

and then marginalize over the bias to obtain vp(indepen-

dent of b) in that redshift bin. Therefore, if our statistical

determination of the history of vi

an independent measurement of bias, then vi

termined precisely. The scaled parameter g∗

on all shift factors; the primordial amplitude or the en-

hancement of Dm due to varying ωm, as well as later

time Θmevolution. But the estimation of vpfrom g∗

is independent of the uncertainty in the overall shifting.

If the evolution of g∗

Θmis measured, it can be used to

reconstruct other coherent growth factors. The coherent

growth factor of Φ can be given using the Euler equation,

biand

pand bifrom the data

pcan be combined with

pcan be de-

Θmidepends

Θmi

g∗

Φ=2

3

aH

H2

∗ωm

?

g∗

θm+dg∗

θm

dlna

?

,(9)

where no anisotropy condition is used. If the Poisson

equation is validated then the re-constructed g∗

used to derive g∗

this estimated matter fluctuation evolution can be used

to determine bias from the measured g∗

Φcan be

Φ. Then

δmusing the relation g∗

δm= ag∗

bias b = g∗

bi/g∗

δm.

III.REDSHIFT–SPACE DISTORTIONS FROM

CLUSTERS OF GALAXIES

As a demonstration of the parameterization discussed

above, we present here a measurement of the redshift–

space 2-D two–point correlation function (ξs(σ,π)) for

clusters of galaxies selected from the SDSS. We use an

updated version of the C4 cluster catalogue [23] based

on Data Release 3 (DR3; [38]) of the SDSS. Briefly, the

C4 catalogue identifies clusters in a seven-dimensional

galaxy position and colour space (righ ascension, declina-

tion, redshift, u−g,g−r,r−i,i−z)using the SDSS Main

Galaxy spectroscopic sample. This method greatly re-

duces the twin problems of projection effects and redshift

space distortions in identifying physically–bound galaxy

groups. This catalogue is composed of ∼ 2000 clusters in

the redshift range 0.02 < z < 0.15.

The estimation of the correlation function relies cru-

cially on our ability to compare the clustering of the data

to that of a random field. Thus any artificial structures

in the data must be considered when constructing the

random catalogue. These problems include incomplete-

ness, such as the angular mask (e.g. survey boundaries,

bright stars and dust extinction in our own galaxy), and

the radial distribution where at large distances, the mean

space density decreases as we approach the magnitude

limit of the survey. We have constructed random sam-

ples which takes these issues into account, i.e., the angu-

lar positions are randomly sampled from a sphere to lie

within the DR3 mask, while the redshifts are obtained

from a smooth spline fit to the real C4 redshift distri-

bution (which removes true large scale structures). The

random samples are then made to be 50 times denser

than the real data to avoid Poisson noise.

In Figure 3, we show our estimation of the ξs(σ,π)

binned into with 6 configuration–space bins up to 60 Mpc

(one bin per 10Mpc). Separations of less than 10 Mpc

are removed to reduce the FoG effect. Error on ξs(σ,π)

were derived using the jackknife method [39], which in-

volves dividing the survey into N sub-sections with equal

area (and thus volume) and then computing the mean

and variance of ξs(σ,π) from these N measurements of

the correlation function with the ithregion removed each

time (where i = 1...N).

In our analysis, we divided the whole C4 area into N =

30 sub-subsections and determine the variance from [40],

σ2

ξ(ri) =Njack− 1

Njack

Njack

?

k=1

[ξk(ri) − ξ(ri)]2,(10)

where Njackis the number of jackknife samples used and

rirepresents a single bin in the σ−π configuration space.

Then we compute

ξ(ri) =

1

Njack

Njack

?

k=1

ξk(ri), (11)

and the normalised covariance matrix is estimated

from [39],

Cij=Njack− 1

Njack

k=Njack

?

k=1

∆k

i∆k

j,(12)

where,

∆k

i=ξk(ri) − ξ(ri)

σξ(ri)

. (13)

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FIG. 4: An ordered list of the Eigenvalues for our cluster covari-

ance matrix.

Before we invert Cijin Eqn. 12, we note that the values

of Cijare estimated to limited resolution,

∆Cij=

?

2

Njk

(14)

and therefore, if Njk is small, or there are degeneracies

within Cij, the inversion will be affected. This problem

can be eliminated by performing a Single Value Decom-

position (SVD) of the matrix,

Cij= U†

ikDklVlj,(15)

where U and V are orthogonal matrices that span the

range and the null space of Cijand Dkl= λ2δkl, a diag-

onal matrix with singular values along the diagonal. In

doing the SVD, we select the dominant modes to con-

tribute to the χ2by requiring that λ2>?2/Njk.

creasing eigenmodes and see a “kink” in the distribu-

tion which we interpret as indicating a transition in the

signal–to–noise of the eigenmodes, i.e., only the first ten

modes contain most of the signal, while higher-ordered

modes are dominated by noise.

eigenmodes beyond this kink (with λi< 0.01) where the

eigenvalues start to flatten out.

In Figure 4, we rank the eigenvalues (λi) for the in-

We therefore remove

A.Statistical determination of large scale flow

As discussed in the Introduction, there is recent evi-

dence for excessive bulk flow motions compared to the

WMAP5-normalised ΛCDM model [18] and therefore, it

is important to confirm these results as it may indicate

evidence for an alternative explanation for the observed

cosmic acceleration such as modified gravity. In this pa-

per, we provide a first demonstration of our new param-

eterization using clusters of galaxies from the SDSS. In

detail, we attempt to model the “squashing” of the 2-

D correlation function of the C4 cluster sample seen in

Figure 5 using the formalism presented herein. We do

however caution the reader that we expect the limited

size of the DR3 sample to leads to large statistical er-

rors, due to a significant shot–noise contribution because

of their low number density. However, future cluster and

galaxy samples (e.g., LRGs) should provide stronger con-

straints and provide a more robust test of these high bulk

flow measurements in the literature.

In Figure 5, we provide the best fit parameters b and

vp for the C4 correlation function presented in Figure

3 and there is as expected a clear anti-correlation be-

tween these two parameters because the anisotropic am-

plitude is generated by cross-correlations in the density

and peculiar velocity fields. The best fit value from Fig-

ure 5 is vp= 270+433km/s (at the 1σ level marginalised

with other parameters including b) and is consistent with

vp= 0. We do not quote the negative bound of the error

on vpas it is below zero and thus has no physical mean-

ing. Instead, we quote the upper bound on vpand note

that our result is consistent with zero. Our measurement

of vp is close to the predicted value of 203 km/s for a

WMAP5–normalised ΛCDM model.

We propose above that vpis a complementary param-

eter for reporting such peculiar velocity measurements.

The parameter gΘ, which is equivalent to fσ8, is not de-

termined precisely without the prior information of AS.

But when we report our measurement with vp, there is no

uncertainty due to other cosmological parameters which

are not determined accurately, as it is equivalent to g∗

determined statistically from redshift space distortion.

The observed value vpat a given redshift is not only in-

dependent of bias but also independent of normalisation.

Θ

B. Reconstruction of matter density field from vp

We convert vp measurement into gΘ using As from

WMAP5 (gΘ: coherent growth factor of peculiar veloc-

ity, and it is equivalent to fσ8 in other parameteriza-

tions). With the evolution of gΘknown, dynamics of per-

turbations are reconstructed to provide the history of Ψ

through the Euler equation. In most theoretical models,

the time variation of vpis minimal at these low redshifts

discussed here for the C4 sample (z ≃ 0.1), which allows

us to ignore the time-derivative part in Eq. 9. Therefore,

it is straightforward to transform the coherent evolution

of Θ into the coherent evolution of Ψ. If we assume no

anisotropic stress, then it is easy to convert to the coher-

ent evolution of Φ, gΦ.

We are able to determine matter density fluctuations

through the Poisson equation. We calculate the coher-

ent growth of δm, gδ = 0.7, which is related to gΦ as

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7

FIG. 5: The 2-D contours between b and vp with the DR3 cluster

sample.

gδ = agΦ, if no modified Poisson equation is assumed.

Finally, the estimated gδcan be used to derive bias using

measured g∗

b. Through fitting to the redshift distortion

effect, we extract both g∗

ation evolution gδ is estimated only from g∗

other measurement g∗

bis not yet used. The combination

of the estimated gδ and the measured g∗

from b = g∗

b/gδ= 2.9±0.8, which is fully consistent with

our expectations for such massive clusters of galaxies in

the C4 sample.

band g∗

Θ. The density fluctu-

Θ, and the

Θprovides bias

IV.DISCUSSION

We outline in this paper a new theoretical model

for ξs(π,σ), the 2-D two–point correlation function in

configuration–space, which allows us to constrain the

bulk flow motion of matter on large scales.

propose that the 1-D linear velocity dispersion (vp) is a

interesting quantity to report when measuring redshift–

space distortions. We demonstrate this method using C4

clusters from the SDSS and find a value for vp that is

consistent with a WMAP5–normalised ΛCDM cosmol-

ogy (within our large statistical errors). Our observed

value for these bulk flows is marginally inconsistent with

other recent observations in the literature, which find an

excess flow compared to a WMAP5–normalised ΛCDM

model [18–20]. We do not discuss this further as we plan

We also

to revisit these measurements using larger datasets and

different tracers of the density field.

As discussed in Section II-C, our measurement of vp

is correlated with gb, which is the combination of b and

δm, since the observed anisotropic shape of the 2-D cor-

relation function in redshift–space is generated by a cross

correlation between the density field and peculiar veloc-

ities. However, one of the important implications of our

method is that we can measure vpwithout knowing how

to decompose gb, and thus without the uncertainty of

determining b.

There are however some caveats to our analysis. We

do not analyse our data in Fourier space, but in config-

uration space. Small scales have been removed from our

data analysis (< 10Mpc), to ensure that the FoG effect

will not contaminate our results. Our methodology is in-

sensitive to a possible shape dependence at large scales

for any exotic reason; scale dependent later time growth

(e.g. f(R) gravity models [41]), or scale–dependent bias

at large scales.In follow-up studies, we will measure

the redshift–space distortions in Fourier space to test the

effect of small–scales on our results. In addition, the for-

mulation to derive ξsused in this paper can be slightly

biased due to the dispersion effect studied in [32]. This ef-

fect is not parameterised properly here, but the reported

level of uncertainty is approximately 5% which is much

smaller than the statistical errors on our present mea-

surements. Therefore, we dismiss this shift here but it is

worth revisiting this issue in the future to know how to

incorporate this effect in a new parameterisation.

Finally, Song and Percival [13] recently proposed a

method to re-construct the structure formation observ-

ables from Θ measurements. Although it is not yet es-

timated precisely in that narrow range of measured val-

ues, we apply their methodology in practice. From the

observed coherent evolution of Θ at z = 0.1 from the

DR3 C4 clusters, we re-construct Ψ, Φ and δm. We then

find that bias can be derived from the estimated δmand

the measured g∗

b. Here, for the first time, we estimate

bias from peculiar velocity measurements only. The esti-

mated values are reasonable at b = g∗

is not precise measurement yet, as the time variation is

ignored, but we will revisit this in a following paper.

b/gδ= 2.9 ± 0.8. It

Acknowledgments

The authors would like to thank Nick Kaiser, Kazuya

Koyama and Will Percival for helpful conversations,

Prina Patel for useful suggestions on the presentation of

this article, and the referee for helpful comments. Y-SS,

RCN and CGS are grateful for support from STFC .

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