# Testing the minimum variance method for estimating large-scale velocity moments

**ABSTRACT** The estimation and analysis of large-scale bulk flow moments of peculiar

velocity surveys is complicated by non-spherical survey geometry, the

non-uniform sampling of the matter velocity field by the survey objects and the

typically large measurement errors of the measured line-of-sight velocities.

Previously, we have developed an optimal `minimum variance' (MV) weighting

scheme for using peculiar velocity data to estimate bulk flow moments for

idealized, dense and isotropic surveys with Gaussian radial distributions, that

avoids many of these complications. These moments are designed to be easy to

interpret and are comparable between surveys. In this paper, we test the

robustness of our MV estimators using numerical simulations. Using MV weights,

we estimate the bulk flow moments for various mock catalogues extracted from

the LasDamas and the Horizon Run numerical simulations and compare these

estimates to the moments calculated directly from the simulation boxes. We show

that the MV estimators are unbiased and negligibly affected by non-linear

flows.

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Page 1

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 28 September 2012(MN LATEX style file v2.2)

Testing the minimum variance method for estimating

large-scale velocity moments

Shankar Agarwal1,?& Hume A. Feldman1,†& Richard Watkins2,‡

1Department of Physics & Astronomy, University of Kansas, Lawrence, KS 66045, USA.

2Department of Physics, Willamette University, Salem, OR 97301, USA.

emails:?sagarwal@ku.edu;†feldman@ku.edu;‡rwatkins@willamette.edu

ABSTRACT

The estimation and analysis of large-scale bulk flow moments of peculiar velocity

surveys is complicated by non-spherical survey geometry, the non-uniform sampling

of the matter velocity field by the survey objects and the typically large measurement

errors of the measured line-of-sight velocities. Previously, we have developed an op-

timal ‘minimum variance’ (MV) weighting scheme for using peculiar velocity data to

estimate bulk flow moments for idealized, dense and isotropic surveys with Gaussian

radial distributions, that avoids many of these complications. These moments are de-

signed to be easy to interpret and are comparable between surveys. In this paper,

we test the robustness of our MV estimators using numerical simulations. Using MV

weights, we estimate the bulk flow moments for various mock catalogues extracted

from the LasDamas and the Horizon Run numerical simulations and compare these

estimates to the moments calculated directly from the simulation boxes. We show that

the MV estimators are unbiased and negligibly affected by non-linear flows.

Key words: galaxies: kinematics and dynamics – galaxies:

statistics – cosmology: observations – cosmology: theory –

distance scale – large scale structure of Universe.

1 INTRODUCTION

Peculiar velocities are a sensitive probe of the underlying

large-scale matter density fluctuations in our Universe. In

particular, large, all-sky surveys of the peculiar velocities of

galaxies or clusters of galaxies can provide important con-

straints on cosmological parameters. However, studies of pe-

culiar velocities suffer from several drawbacks, including (i)

the presence of small-scale, non-linear flows, such as infall

into clusters, can potentially bias analyses which typically

rely on linear theory, (ii) sparse, non-uniform sampling of

the peculiar velocity field can lead to aliasing of small-scale

power on to large scales and bias due to heavier sampling of

dense regions, (iii) large measurement uncertainties of indi-

vidual peculiar velocity measurements, particularly for dis-

tant galaxies or clusters, make it necessary to work with

large surveys in order to extract meaningful constraints.

These difficulties have often been addressed by calculat-

ing statistics from peculiar velocity surveys that are designed

to primarily reflect large-scale flows which are well described

by linear theory. The most common statistic used is the bulk

flow, which represents the average motion of the objects in a

survey. The bulk flow statistic has been investigated exten-

sively by many groups (Dressler & Faber 1990; Kaiser 1991;

Feldman & Watkins 1994; Jaffe & Kaiser 1995; Strauss et al.

1995; Watkins & Feldman 1995; Hudson et al. 1999, 2004; da

Costa et al. 2000a; Parnovsky & Tugay 2004; Sarkar, Feld-

man, & Watkins 2007; Kashlinsky et al. 2008, 2010; Ma,

Gordon, & Feldman 2011; Macaulay et al. 2011; Nusser,

Branchini, & Davis 2011; Nusser & Davis 2011; Abate &

Feldman 2012; Turnbull et al. 2012). However, bulk flow es-

timates can be difficult to interpret since how they sample

the peculiar velocity field depends strongly on the character-

istics of the particular survey being considered. In addition,

results from bulk flow analyses have often been controversial,

highlighting the importance of developing a robust bulk flow

statistic that is easy to interpret and that can be compared

between surveys with different geometries.

In Watkins, Feldman, & Hudson 2009 (hereafter Paper

I) and Feldman, Watkins, & Hudson 2010 (hereafter Paper

II), we developed the ‘minimum variance’ (MV) moments

that were designed to estimate the bulk flow of a volume

of a given scale rather than a particular peculiar velocity

survey. We stress that the MV moments do not represent

the bulk motion of the galaxies in a survey, rather they are

estimates of the bulk motion of a given volume of space. The

MV algorithm was designed to make a clean estimate of the

large-scale bulk flow as a function of scale using the avail-

able peculiar velocity data. Essentially, each velocity datum

c ? 0000 RAS

arXiv:1201.0128v3 [astro-ph.CO] 27 Sep 2012

Page 2

2Agarwal & Feldman & Watkins

in a real survey is weighted in a way that minimizes the vari-

ance of the difference between the MV-weighted bulk flow

of the real survey and an idealized survey bulk flow, on a

characteristic scale R. The MV analysis suggested bulk flow

velocities well in excess of expectations from a Λ cold dark

matter (ΛCDM) model with 7-year Wilkinson Microwave

Anisotropy Probe (WMAP7; Larson et al. 2011) central pa-

rameters.

Indeed there are a few recent observations that sug-

gest that the standard model may be incomplete: large-scale

anomalies found in the maps of temperature anisotropies in

the cosmic microwave background (CMB; Copi et al. 2010;

Sarkar et al. 2011; Bennett et al. 2011); a recent estimate

(Lee & Komatsu 2010) of the occurrence of high-velocity

merging systems such as the bullet cluster is unlikely at a

∼6σ level; large excess of power in the statistical clustering of

luminous red galaxies (LRG) in the photometric Sloan Dig-

ital Sky Survey (SDSS) galaxy sample (Thomas, Abdalla,

& Lahav 2011); Kovetz, Ben-David, & Itzhaki (2010) find

a unique direction in the CMB sky determined by anoma-

lous mean temperature ring profiles, also centred about the

direction of the flow detected above; larger than expected

cross-correlation between samples of galaxies and lensing

of the CMB (Ho et al. 2008; Hirata et al. 2008); Type Ia

Supernovae (SNIa) seem to be brighter than expected at

high redshift (Kowalski et al. 2008); small voids (∼ 10 Mpc)

are observed to be much emptier than predicted (Gottl¨ ober

et al. 2003); observations indicate denser high concentration

cluster haloes than the shallow low concentration and den-

sity profile predictions (de Blok 2005; Gentile et al. 2005).

In this paper, we use N-body simulations to investigate

the robustness of our MV scheme for estimating the bulk

flow moments of the velocity field, over a volume of a partic-

ular scale, R. First we extract a mock catalogue (described in

Sec. 3) from N-body simulations. Given this mock catalogue,

we use our MV algorithm (described in Sec. 2) to estimate

the bulk flow moments {ux,uy,uz} of the velocity field over

a volume of a particular scale. Then we position ourselves

in the N-body simulation box at the location of the centre

of the mock catalogue, and calculate the Gaussian-weighted

moments {Vx,Vy,Vz} by averaging the velocities of all the

galaxies in the simulation box; each galaxy being weighted

by a Gaussian radial distribution function f(r) = e−r2/2R2.

Note that a large number of particles in the simulation box

is preferable to accurately calculate the Gaussian moments

of the velocity field. Finally, we compare the MV-weighted

moments {ux,uy,uz} with the Gaussian-weighted moments

{Vx,Vy,Vz} in Sec. 4. A close match between the two would

indicate that the MV scheme accurately estimates the Gaus-

sian bulk flow on scale R.

It is worth mentioning here the reason for our choice

of a Gaussian profile f(r) over, for example, a Tophat filter

in developing the MV formalism. A Tophat filter gets con-

tribution from small scales. As such, bulk flow calculated

using a Tophat filter can be compared with theoretical ex-

pectations only if the observed velocity field is reasonably

dense and uniform, so that the small-scale systematics av-

erage out. However, observations typically are sparse and

non-uniform with large uncertainties. This leads to aliasing

of small-scale power on to large scales, making comparison

with theory difficult. A Gaussian filter, on the other hand,

gets very little contribution from small scales and isolates

the small-scale effects present in real surveys, thereby mak-

ing comparison with theoretical predictions meaningful. Our

MV method is specifically designed to convert the observed

velocity field into a Gaussian field on a user-specified scale

R.

In Sec. 2 we review the MV formalism. In Sec. 3 we

describe the simulations we use and surveys we model to

extract the mock catalogues. In sec. 4 we compare the MV-

weighted bulk flow moments with the Gaussian-weighted

moments. We discuss our results and conclude in Sec. 5.

2 THE MINIMUM VARIANCE METHOD

Individual

plagued by large uncertainties and contributions from small-

scale, non-linear processes which are difficult to model the-

oretically. Both of these problems can be greatly reduced if

instead of considering individual velocities an average veloc-

ity over a sample, commonly called the bulk flow, is worked

with. The three components of the bulk flow ui can be writ-

ten as weighted averages of the measured radial peculiar

velocities of a survey,

?

where Sn is the radial peculiar velocity of the nth galaxy

of a survey, and wi,n is the weight assigned to this veloc-

ity in the calculation of ui. Throughout this paper, sub-

scripts i,j and k run over the three components of the bulk

flow, while subscripts m and n run over the galaxies. By

far the most common weighting scheme used in studies of

the bulk flow, which we will call the maximum likelihood

estimate (MLE) method, is obtained from a maximum like-

lihood analysis introduced by Kaiser (1988). By modelling

galaxy motions as being due to a uniform flow and assuming

Gaussian-distributed measurement uncertainties, the likeli-

hood function

radialpeculiarvelocitymeasurementsare

ui =

n

wi,nSn, (1)

L[ui|{Sn,σn,σ∗}] =

?

n

1

?

σ2

n+ σ2

∗

exp

?

−1

2(Sn− ˆ rn,iui)2

σ2

n+ σ2

∗

?

.

(2)

is obtained, where ˆ rn is the unit position vector of the nth

galaxy, σnis the measurement uncertainty of the nth galaxy

and σ∗ is a 1D velocity dispersion accounting for smaller

scale motions. Maximizing this likelihood gives a bulk flow

estimate of the form of Eq. 1, with weights

wi,n =

3

?

j=1

A−1

ij

ˆ rn,j

σ2

n+ σ2

∗, (3)

where

Aij =

?

n

ˆ rn,iˆ rn,j

σ2

n+ σ2

∗.(4)

These weights play the dual roles of accounting for geomet-

rical factors, e.g. picking out the x component of velocities

c ? 0000 RAS, MNRAS 000, 000–000

Page 3

Minimum variance velocity moments3

in a calculation of ux, and down-weighting velocities with

large uncertainties. However, the fact that velocity uncer-

tainties are typically proportional to distance, together with

the sparseness of velocity catalogues at their outer edges,

means that nearby objects are greatly emphasized in calcu-

lations of the MLE bulk flow. Indeed, studies of the window

functions of these moments (Paper I) have shown that MLE

bulk flow moments of a survey are typically sensitive to flows

on scales much smaller than the survey’s physical diameter,

thus complicating their interpretation.

In Paper I, we introduced an alternative to the MLE

weights that yield bulk flow moments that are much easier

to interpret. First, we imagine an idealized survey contain-

ing radial velocities that well sample the velocity field in a

region. This survey consists of a large number of objects,

all with zero measurement uncertainty. For simplicity, the

radial distribution of this idealized survey is taken to be a

Gaussian profile of the form f(r) ∝ e−r2/2R2, where R gives

a measure of the depth of the survey. This idealized sur-

vey has easily interpretable bulk flow components Ui that

are not affected by small-scale aliasing and which reflect the

motion of a well-defined volume. Note that the difference

between Ui and Vi (see Sec. 1 for the definition of Vi) is that

Ui is calculated from an ideal (dense and isotropic) survey,

while Vi is based on the galaxy distribution obtained from

N-body simulations. In the limit that the simulations are

dense enough, Vi will converge towards Ui.

Our goal is to construct estimators for the idealized

survey bulk flow components Ui, out of the measured radial

peculiar velocities Sn and positions rn contained in a real

survey. We assume that Sncan be expressed as Sn = vn+δn,

where vnis the radial component of the linear peculiar veloc-

ity field at the location of the object and δn accounts for the

measurement noise as well as any non-linear flow, e.g. infall

into a cluster. In order to calculate the weights to use for the

bulk flow estimators, we minimize the variance ?(ui−Ui)2?,

where the average is over different realizations of a particu-

lar matter power spectrum. Expanding this expression using

Eq. 1 for the bulk flow estimate, we obtain

?(ui− Ui)2?

=

?

−2

m,n

wi,mwi,n?SmSn? + ?U2

?

i?

(5)

n

wi,n?Uivn?,

where we have used the fact that the measurement error

included in Sn is uncorrelated with the bulk flow Ui.

Before we minimize this expression with respect to the

weights wi,n, we impose the following constraint introduced

in Paper II. Suppose that the velocity field were a pure bulk

flow, so that Sn =?

of the nth galaxy {ˆ rn,x, ˆ rn,y, ˆ rn,z} and δn is the noise due to

measurement error. We ask that the estimators ui give the

correct amplitude for the flow on average (over different re-

alizations of the universe), namely that ?ui? = Ui. Plugging

the expression for Sn into Eq. 1 give the constraint that

?

iUigi(rn)+δn, where Ui are the three

bulk moments {Ux,Uy,Uz}; gi(rn) are the direction cosines

n

wi,ngj(rn) = δij,(6)

δij being the Kronecker delta. This set of three con-

straints is implemented using Lagrange multipliers, so that

we derive the desired weights by taking a derivative of the

expression

?

3

?

with respect to wi,n and setting the resulting expression

equal to zero. Solving for the weights then gives

?

m,n

wi,mwi,n?SmSn? + ?U2

??

i? − 2

?

?

n

wi,n?Uivn?

(7)

+

j=1

λij

n

wi,ngj(rn) − δij

wi,n =

?

m

G−1

mn

?SmUi? −1

2

3

?

j=1

λijgj(rm)

?

,(8)

where G is the covariance matrix of the individual measured

velocities, Gmn = ?SmSn?. The Lagrange multipliers can be

found by plugging Eq. 8 into Eq. 6 and solving for λij,

?

m,n

λij =

3

?

k=1

M−1

ik

??

G−1

mn?SmUk?gj(rn) − δjk

??

,(9)

where the matrix M is given by

Mij =1

2

?

m,n

G−1

mngi(rn)gj(rm).(10)

In linear theory, the correlation ?SmUi? and the covari-

ance matrix G that appear in our expression for wi,n can

be calculated for a given matter power spectrum P(k) (for

details see Paper II):

?SmUi?

=

N?

?

N?

?

n?=1

w?

i,n??Smvn??

(11)

=

n?=1

w?

i,n?H2

0Ω1.1

2π2

m

?

dk P(k)fmn?(k),

where

w?

i,n? =

3

?

j=1

A−1

ij

ˆ r?

n?,j

N?

are the weights of an ideal, isotropic survey consisting of

N?exact radial velocities vn? measured at randomly selected

positions r?

n? with

Aij =

N?

?

n?=1

ˆ r?

n?,iˆ r?

N?

n?,j

,

Gmn

=

H2

0Ω1.1

2π2

?ˆ rn· v(rn) ˆ rm· v(rm)? + δmn(σ2

where fmn(k) is the angle-averaged window function:

?

×exp

m

?

dk P(k)fmn(k) + δmn(σ2

∗+ σ2

n) (12)

=

∗+ σ2

n),

fmn(k)=

d2ˆk

4π

?

ikˆk · (rm− rn)

ˆ rm·ˆk

??

ˆ rn·ˆk

?

?

(13)

?

.

c ? 0000 RAS, MNRAS 000, 000–000

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4Agarwal & Feldman & Watkins

DEEP-survey

DEEP-mock

Figure 1. Top row: DEEP catalogue (left) and its radial distri-

bution (right). Bottom row: DEEP mock catalogue (left) and its

radial distribution (right).

Thus, given a peculiar velocity survey and a power spec-

trum model P(k) we can calculate the optimum weights wi,n

(see Eq. 8) for estimating the MV moments (see Eq. 1). We

use the power spectrum model given by Eisenstein & Hu

(1998) with WMAP7 (Larson et al. 2011) central parame-

ters. Using the optimum weights wi,n from Eq. 8, the angle-

averaged tensor window function W2

(for details see Paper II) as

?

The diagonal elements W2

the bulk flow components ui. Given a velocity survey, W2

estimated using the MV weights are the closest approxima-

tion to the ideal window functions. See Paper I for the MV-

estimated window functions of the bulk flow components for

a range of surveys.

ij(k) can be constructed

W2

ij(k)=

m,n

wi,mwj,nfmn(k).(14)

iiare the window functions of

ij

3MOCK CATALOGUES

3.1N-body simulations

To check the robustness of our MV formalism, we calcu-

lated the bulk flow moments directly from numerical simu-

lations. The N-body simulations we use in our analysis are

(i) the Large Suite of Dark Matter Simulations (LasDamas;

hereafter LD; McBride et al. 2009; McBride et al. 2011,

in prep1) and (ii) Horizon Run (hereafter HR; Kim et al.

2009). These are designed to model the SDSS observations.

1http://lss.phy.vanderbilt.edu/lasdamas/download.html

COMPOSITE-survey

COMPOSITE-mock

Figure 2. Top row: COMPOSITE catalogue (left) and its radial

distribution (right). Bottom row: COMPOSITE mock catalogue

(left) and its radial distribution (right). The mock does not have

as many close by objects as there are in the COMPOSITE cata-

logue.

The LD (HR) simulation parameters are Ωm = 0.25 (0.26),

Ωb = 0.04 (0.044), ΩΛ = 0.75 (0.74), h = 0.7 (0.72), σ8 =

0.8 (0.794), ns = 1.0 (0.96) and LBox = 1 (6.592)h−1Gpc

for the matter, baryonic and cosmological constant normal-

ized densities, the Hubble parameter, the amplitude of mat-

ter density fluctuations, the primordial scalar spectral index

and the simulation box size, respectively. The HR simulation

samples the density field at z = 0 and identifies galaxies us-

ing subhalos (Kim, Park, & Choi 2008). The LD simulations,

a suite of 41 independent realizations of dark matter N-body

simulations named Carmen, have information at z = 0.13.

Using the Ntropy framework (Gardner et al. 2007), bound

groups of dark matter particles (halos) are identified with

a parallel friends-of-friends (FOF) code (Davis et al. 1985).

The cosmological parameters and the design specifications

of the LD-Carmen and HR simulations are listed in Table 1.

The LD-Carmen data we use consists of 41 independent

realizations, each in a 1h−1Gpc box with the same initial

power spectrum, but a different random seed. We extract

100 mock catalogues from each of the 41 LD boxes, for a

total of 4100 mocks. The mock centres are randomly chosen

inside the box. The mocks are extracted in a way that they

come as close as possible to the radial distribution of real

catalogues. The HR simulation is a single realization in a

much bigger 6.592h−1Gpc box. As such, we extract 5000

randomly distributed mocks.

c ? 0000 RAS, MNRAS 000, 000–000

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Minimum variance velocity moments5

Table 1. The cosmological parameters and the design specifications of the LD-Carmen and HR simulations.

LD-Carmen HR

Cosmological parameters

Matter density, Ωm

Cosmological constant density, ΩΛ

Baryon density, Ωb

Hubble parameter, h (100 km s−1Mpc−1)

Amplitude of matter density fluctuations, σ8

Primordial scalar spectral index, ns

0.25

0.75

0.04

0.7

0.8

1.0

0.26

0.74

0.044

0.72

0.794

0.96

Simulation design parameters

Simulation box size on a side (h−1Mpc)

Number of CDM particles

Initial redshift, z

Particle mass, mp (1010h−1M?)

Gravitational force softening length, f? (h−1kpc)

1000

11203

49

4.938

53

6592

41203

23

29.6

160

Figure 3. The left-hand panel shows the distribution of galaxies around the location of the centre of a typical mock catalogue. Each

galaxy is weighted with a Gaussian radial distribution function f(r) = e−r2/2R2(here R = 50 h−1Mpc). The radial distribution is shown

in the right-hand panel. The MV formalism estimates the bulk flow of this Gaussian-weighted box, by only using the mock catalogues of

the kind shown in Figs 1 and 2 (bottom rows).

3.2Catalogues

We create mocks of three different peculiar velocity surveys

from the simulations: i) The ‘DEEP’ compilation includes

103 SNIa (Tonry et al. 2003), 70 Spiral Galaxy Culsters (SC)

Tully-Fisher (TF) clusters (Giovanelli et al. 1998; Dale et al.

1999a), 56 Streaming Motions of Abell Clusters (SMAC)

fundamental plane (FP) clusters (Hudson et al. 1999, 2004),

50 Early-type Far Galaxies (EFAR) FP clusters (Colless

et al. 2001) and 15 TF clusters (Willick 1999). The DEEP

catalogue consists of 294 data points with a characteristic

MLE depth of 50 h−1Mpc, calculated using?wnrn/?wn

we assume σ∗ = 150 km s−1. We have tried σ∗ = 150 − 450

km s−1and it does not change our results appreciably. ii)

where the MLE weights are wn = 1/(σ2

n+σ2

∗). In this paper,

The SFI++ (Spiral Field I-band) catalogue (Masters et al.

2006; Springob et al. 2007, 2009) is the densest and most

complete peculiar velocity survey of field spirals to date.

We use the data from the corrected dataset (Springob et al.

2009), the sample consists of 2821 TF field galaxies. The

characteristic depth is 34 h−1Mpc. iii) The ‘COMPOSITE’

catalogue is a compilation of the DEEP and SFI++ cata-

logues as well as the group SFI++ catalogue (Springob et al.

2009), the Early-type Nearby Galaxies (ENEAR; da Costa

et al. 2000b; Bernardi et al. 2002; Wegner et al. 2003) survey

and a surface brightness fluctuations (SBF) survey (Tonry

et al. 2001). With 4481 data points, the COMPOSITE cat-

alogue has a characteristic depth of 33 h−1Mpc. The DEEP

and SFI++ catalogues are completely independent whereas

the COMPOSITE is a compilation of these and other cata-

c ? 0000 RAS, MNRAS 000, 000–000

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6Agarwal & Feldman & Watkins

logues. For further details on these catalogues see Papers I

and II.

We have used these particular catalogues to investigate

the effect of geometry and density on our results. The rea-

son for using these catalogues is that we want to compare

the results using a very sparse catalogue (DEEP) and the

better sky coverage and higher density of the COMPOSITE

catalogue. We chose the SFI++ catalogue as an intermedi-

ate case study. We tested our MV formalism on the DEEP,

SFI++ and COMPOSITE mocks extracted from the LD

and HR simulations. As we mentioned earlier, we extracted

4100 mocks from the LD simulations and 5000 from the

HR simulation. The results based on the 5000 mock surveys

from the HR simulation are virtually identical to the ones

for the LD simulations. As such, in the rest of this paper, we

display results only for the 4100 mocks extracted from the

LD simulations. Moreover, since our results for the SFI++

catalogue are very similar to the ones for the DEEP and

COMPOSITE catalogues, we do not display SFI++ results.

In Figs 1 and 2, we show the DEEP and COMPOS-

ITE real catalogues (top rows) and a sample mock cata-

logue (bottom rows). The N-body simulations do not have

as many close by objects as there are in the COMPOSITE

catalogue, which is why the COMPOSITE mocks match the

radial distribution only beyond ∼ 50h−1Mpc.

In Fig. 3, we show the weighted distribution of galaxies

around the location of the centre of a typical mock cat-

alogue (left-hand panel) and its radial distribution (right-

hand panel). Each galaxy is weighted with a Gaussian radial

distribution function f(r) = e−r2/2R2with R = 50h−1Mpc.

The MV formalism is designed to obtain the best estimate

of the bulk flow of this Gaussian-weighted box, by only us-

ing the mock catalogues of the kind shown in Figs 1 and 2

(bottom rows). Note that the Gaussian-weighted box does

not have a perfect Gaussian distribution but it comes close

to being one. Denser simulations would be required to test

the MV formalism more rigorously.

3.3Mock extraction procedure

Once we have identified a random point in the N-body sim-

ulation box, we extract a set of galaxies that has the same

radial selection function about this point as the catalogue we

are creating mocks of. We do not impose the additional con-

straint on the mocks that they must also have the same an-

gular distribution as the real surveys for two reasons: (i) the

N-body simulations are not dense enough to give us mocks

that are exactly like the real surveys and (ii) the weights

wi,n of the real surveys typically depend only on the radial

distribution and the velocity errors of the survey objects.

Consequently, the mocks in Figs 1 and 2 have a relatively

featureless angular distribution. To make the mocks more

realistic, we also impose a 10olatitude zone-of-avoidance

cut.

From the simulations we find the angular position,

the true line-of-sight peculiar velocity vs and the redshift

cz = ds+ vs for each mock galaxy, where ds is the true ra-

dial distance of the mock galaxy from the random centre we

selected, all in km s−1. We then perturb the true radial dis-

Figure 4. Histograms showing the normalized probability dis-

tribution for the MV- and Gaussian-weighted bulk flow moments

within a Gaussian window of radius R = 50 h−1Mpc for the direc-

tions x and z in the top and bottom rows, respectively for the two

types of mock catalogues in the LD simulations: DEEP (left-hand

column) and COMPOSITE (right-hand column) as in Fig. 6. The

MV-weighted bulk flow moments uiare the solid histogram. The

Gaussian-weighted moments Vi are shown as dashed histogram.

We also superimpose a Gaussian centred at zero with width of

the rms calculated. It is clear that the distributions of both the

MV- and Gaussian-weighted moments are Gaussian distributed.

We do not show the y-direction since it is statistically identical to

the x-direction. The SFI++ catalogue shows very similar trends

and so was not displayed.

tance dsof the mock galaxy with a velocity error drawn from

a Gaussian distribution of width equal to the corresponding

real galaxy’s velocity error, σn. Thus, dp = ds+δd, where dp

is the perturbed radial distance of the mock galaxy (in km

s−1) and δd is the velocity error. The mock galaxy’s mea-

sured line-of-sight peculiar velocity vp is then assigned to

be vp = cz − dp, where cz is the redshift we found above.

The reason for this procedure is that the weight we assign

to each galaxy in the mock catalogues will then be similar

to the weights of the real catalogues, since these depend on

the radial distribution errors of the survey objects.

This procedure of perturbing the distances ds and then

assigning the velocities vp to the mock galaxies introduces

a Malmquist bias. We have checked the effect of the bias by

following a slightly different approach to generate the mocks.

We used the exact distances ds and only perturbed the ve-

locities as vp = vs+ δd. We found the effect of Malmquist

bias on our MV analyses to be negligible.

c ? 0000 RAS, MNRAS 000, 000–000

Page 7

Minimum variance velocity moments7

Figure 5. The MV bulk flow moments uiversus the Gaussian-

weighted moments Vi for R = 50 h−1Mpc for the two types of

mock catalogues in the LD simulations: DEEP (left-hand column)

and COMPOSITE (right-hand column). There are 4100 mocks for

each of the catalogues. We show the moments ux and uz in the

top and bottom rows, respectively. The MV- and the Gaussian-

weighted moments are plotted against each other (dots). A perfect

correlation would put all 4100 dots on the diagonal. The rms

scatter (km s−1) in the MV moments is displayed at the top

left-hand side of each panel and shown as dashed lines. We do

not show the y-direction since it is statistically identical to the

x-direction. The SFI++ catalogue shows very similar trends and

so was not displayed.

4 BULK FLOW MOMENTS

For each of the 4100 LD (5000 HR) mocks, we estimated

the bulk flow moments {ux,uy,uz} using our MV weight-

ing scheme (Sec. 2). We then compared the results to the

Gaussian-weighted bulk moments {Vx,Vy,Vz} calculated by

going to the same central points for each of the 4100 LD

(5000 HR) mock catalogues and averaging the velocities of

all the galaxies in the simulation box, each galaxy being

weighted by a Gaussian weight of width R = 50h−1Mpc.

Although the results that we show here are for a particular

scale of R = 50h−1Mpc, we have repeated our analysis for

other values of R with similar results. It is worth mentioning

here that since the position and the velocity of every galaxy

in the N-body simulations are known exactly, their respec-

tive uncertainties are zero. Here we present our results only

from the LD simulations. The HR simulation shows very

similar results.

In Fig. 4, we show the probability distribution for the

the 4100 MV-weighted bulk flow moments ui(solid) and the

Gaussian-weighted moments Vi (dashed) within a Gaussian

window of radius R = 50 h−1Mpc for the LD simulations. As

shown in Fig. 4, the distributions for the MV-estimated bulk

Figure 6. Histograms showing the normalized probability dis-

tribution for the differences between the MV- and Gaussian-

weighted moments for the x- and z- directions in the top and

bottom rows, respectively. The solid histograms show the quanti-

ties (ui−Vi) for the 4100 mock catalogues extracted from the 41

LD simulation boxes for R = 50 h−1Mpc: DEEP (left-hand col-

umn) and COMPOSITE (right-hand column). Superimposed on

the histograms are Gaussians centred at zero and with the same

width, ?(ui− Vi)2?

that the distributions are centred on zero demonstrates that the

MV estimators are not biased. We do not show the y-direction

since it is statistically identical to the x-direction. The SFI++

catalogue shows very similar trends and so was not displayed.

1

2, as the corresponding histogram. The fact

flow moments (solid histogram) and the Gaussian-weighted

moments (dashed histogram) are both Gaussian distributed.

This is as expected for large scale moments and reflects

the fact that non-linear motions, which can lead to non-

Gaussian tails in the velocity distributions for individual

galaxies, have been effectively averaged out. The widths of

the distributions match well with the expectations from lin-

ear theory,

σ2

v(R)=

H2

0Ω1.1

2π2

m

?

dk P(k)W2

v(kR),(15)

where σv(R) is the RMS value of the peculiar velocity

field smoothed with a suitable filter with a characteristic

scale R; Wv(k,R) is the window function (Fourier trans-

form of the filter) and P(k) is the matter power spectrum.

A ΛCDM model with WMAP7 (Larson et al. 2011) cen-

tral parameters, together with a Gaussian window function

Wv(k,R) = e−(kR)2/2, predicts a 110 km s−1width for

R = 50 h−1Mpc, virtually identical to the ones shown in

Fig. 4. In Paper II we estimated that for a ΛCDM model

with WMAP7 central parameters, the chance of getting a

c ? 0000 RAS, MNRAS 000, 000–000

Page 8

8Agarwal & Feldman & Watkins

∼ 400 km s−1bulk flow for a survey on scales of 50 h−1Mpc

is ∼ 1 per cent. Examining Fig. 4 confirms that the proba-

bility will be similarly small. Indeed, the frequency of mock

catalogues with > 400 km s−1was found to be comparable

to the 1 per cent value.

In Fig. 5, we show the bulk flow moments in the x-

and z- directions (in the top and bottom rows, respectively)

for the 4100 DEEP (left-hand column) and COMPOSITE

(right-hand column) mock catalogues, extracted from the

41 LD simulation boxes. The MV-weighted moments ui and

the corresponding Gaussian-weighted moments Vi are plot-

ted against each other (dots) and the positive correlation be-

tween the two is clearly visible. A perfect correlation would

put all 4100 points on the diagonal.

In Fig. 6, we show the probability distribution for the

difference between the MV-weighted bulk flow moments ui

and the Gaussian-weighted ideal moments Vi for the 4100

mock surveys from the LD simulations. A Gaussian centred

at zero and with the same width as the probability distri-

bution is also shown. The fact that the distributions are

centred on zero demonstrates that the MV estimators are

not biased.

Given a mock catalogue, the theoretical expectation

value for the width of the distribution, i.e. ?(ui−Ui)2?

be calculated in linear theory using Eq. 5, Eq. 11, Eq. 12 and

Eq. 13. To check the robustness of our MV method, this can

then be compared with the distribution width ?(ui−Vi)2?

calculated directly from the simulations [the (ui− Vi) dis-

tribution is shown in Fig. 6] using the same cosmological

model. The theoretical widths ?(ui − Ui)2?

LD mocks are shown in Table 2, columns 7 – 9. Since each

mock catalogue has in principle a slightly different expec-

tation for the width, we quote the average and standard

deviation of the widths obtained from the set of the mock

catalogues. The widths ?(ui− Vi)2?

tions are shown in Table 2, columns 4 – 6.

Comparing linear theory predictions [?(ui− Ui)2?

Table 2, columns 7 – 9] with the widths found in the numer-

ical simulations [?(ui− Vi)2?

the distribution width found in the simulations are some-

what different than the widths predicted by linear theory.

This is due to the fact that the simulations are not dense

enough and thus do not have enough galaxies to emulate

an ideal survey. We explain this through Fig. 7. In the left-

hand panel, we show the weighted distribution of galaxies

around the location of the centre of a typical mock cata-

logue. Each galaxy is weighted with a Gaussian radial dis-

tribution function f(r) = e−r2/2R2

The right-hand panel shows the window functions W2

bulk flow components ui for this distribution (dash-dotted,

short-dashed and long-dashed lines for the x,y and z com-

ponents, respectively) and the ideal window function (solid

line). Non-Gaussianity in the distribution of galaxies in the

left-hand panel causes a slight mismatch between its win-

dow functions and the ideal one. With a larger number of

galaxies in the simulations, the Gaussian-weighted moments

Vi would approach the ideal moments Ui, and give a closer

match between ?(ui−Ui)2?

1

2, can

1

2

1

2 for the 4100

1

2 found in the simula-

1

2 in

1

2, columns 4 – 6], we see that

with R = 50h−1Mpc.

iiof the

1

2 and ?(ui−Vi)2?

1

2. The DEEP

catalogue does not have as many close by galaxies as in the

SFI++ and COMPOSITE catalogues, and thus the variance

estimates calculated using linear theory [?(ui− Ui)2?

the LD and HR simulations [?(ui− Vi)2?

closer to each other. Taken together with the lack of bias (see

Fig. 6), it is clear that non-linear motions are not having a

significant effect on these large-scale moments.

1

2] and

1

2] are significantly

The much improved performance of MV formalism over

the widely used MLE scheme is also evident in Fig. 8, where

we show the window functions W2

nents, calculated using MV (thick) and MLE (thin) meth-

ods. These window functions correspond to the DEEP (left-

hand column) and COMPOSITE (right-hand column) real

catalogues, for R = 20 h−1Mpc (top row) and R = 50

h−1Mpc (bottom row). For both DEEP and COMPOS-

ITE catalogues, the MV window functions are a reasonable

match to the ideal ones. The MLE window functions are not

only contaminated by small-scale power, they are also very

different for the x-, y- and z-directions – making it difficult

to interpret the MLE bulk flow moments. On the other hand,

by directly controlling the survey window functions the MV

formalism effectively suppresses the small-scale contribution

to the bulk flow. Since it is the small-scales that are predom-

inantly plagued by non-linear effects, the MV scheme is able

to make a clean estimate (compared to MLE) of the bulk

flow components, while keeping the non-linear contamina-

tion to a minimum.

iiof the bulk flow compo-

In Table 2, columns 1 – 3, we also show the values of

the theoretical widths ?(ui−Ui)2?

on which the mocks are based. We see that the theoretical

widths for the real catalogues (columns 1 – 3) are somewhat

larger than the theoretical widths for the mocks (columns 7

– 9). This is due primarily to the fact that the objects in the

simulated catalogues are less clumped than in the real cat-

alogues, even though they have similar radial distribution

functions. This is evident in Figs 1 and 2, where the mock

catalogues can be seen as having a relatively featureless spa-

tial distribution. Less clumping and fewer close by galaxies

in the simulations lower the MV-weighted bulk flow mo-

ments ui, resulting in somewhat lower widths ?(ui−Ui)2?

than the real catalogue widths. The creation of mock cata-

logues with widths that more closely matched the real cat-

alogue widths would require simulations with higher resolu-

tion.

1

2 from the real catalogues

1

2

We also found that the sparser the mock catalogue is

(eg. DEEP), the higher the chances of getting large velocities

(see the extended tails in the velocity distributions for the

DEEP mocks in Fig. 4), but in a way that is consistent with

the larger uncertainties associated with the estimators de-

rived from these mock catalogues. This can be seen by com-

paring the predicted distribution widths ?(ui − Ui)2?

the DEEP and COMPOSITE mock catalogues in Table 2,

columns 7 – 9. The DEEP mocks, being sparser compared

to the COMPOSITE mocks, have larger widths. Compar-

ing the widths of (ui − Vi) histograms (Table 2, columns

4 – 6) found in the simulations (Fig. 6), we again see that

the DEEP mocks have marginally larger uncertainties in the

bulk estimators, as expected.

1

2 for

c ? 0000 RAS, MNRAS 000, 000–000

Page 9

Minimum variance velocity moments9

Table 2. The theoretical distribution width ?(ui− Ui)2?

calculated in linear theory using Eq. 5, Eq. 11, Eq. 12 and Eq. 13. In the fourth (x), fifth (y) and sixth (z) columns, we show the

widths ?(ui− Vi)2?

theoretical widths for the LD mocks are shown in the seventh (x), eighth (y) and ninth (z) columns. For the LD mocks, we quote the

mean and standard deviation values of ?(ui− Ui)2?

2011) central power spectrum parameters. In the last column we show the width of the distribution of the moments ui over the 4100

mock catalogues (see Fig. 4). Since the widths ux,uy and uz were all found to be very similar, we only quote a single value for uiin the

last column. All values are in km s−1.

1

2 for the real catalogues in the first (x), second (y) and third (z) columns,

1

2 of the (ui− Vi) histograms for the LD mocks (see Fig. 6), this should be compared to the first three columns. The

1

2, for the 4100 mock catalogues. These values are based on WMAP7 (Larson et al.

Real catalogues

?(ui− Ui)2?

73.2880.27

57.3758.63

46.7547.34

LD mock catalogues

1

2

?(ui− Vi)2?

73.79

72.38

70.69

1

2

?(ui− Ui)2?

65.37 ± 2.79

56.96 ± 2.58

42.59 ± 1.60

1

2

Width

DEEP

SFI++

COMPOSITE

54.21

47.42

34.93

74.95

73.48

71.47

69.62

72.60

71.34

65.16 ± 2.71

56.52 ± 2.53

42.16 ± 2.15

57.32 ± 1.98

51.37 ± 2.20

39.87 ± 1.17

111

111

112

Figure 7. Left: the distribution of galaxies around the location of the centre of a typical mock catalogue. Each galaxy is weighted with

a Gaussian radial distribution function f(r) = e−r2/2R2(here R = 50 h−1Mpc). Right: The window functions W2

bulk flow components ui, for R = 50 h−1Mpc. The x,y and z components are dash-dotted, short-dashed, long-dashed lines, respectively,

and correspond to the distribution in the left-hand panel. The solid line is the ideal window function (since the ideal survey is isotropic,

all components are the same).

ii(see Eq. 14) of the

5DISCUSSION AND CONCLUSIONS

In previous papers (Papers I and II), we developed a weight-

ing scheme for analyzing peculiar velocity surveys that gives

estimators of idealized bulk flow moments that reflect the

flow of a volume of a particular scale centred on our loca-

tion rather than the characteristics of a particular survey.

Given a peculiar velocity survey, the MV method is capa-

ble of ‘redesigning’ the survey window function in a way

that minimizes the aliasing of small-scale power on to large

scales, thereby making comparisons with linear theory as

well as among independent surveys possible. The direct con-

trol over a survey window function makes the MV formalism

an extremely useful tool when comparing bulk flow results

across independent surveys with varying characteristics.

Using mock catalogues drawn from numerical simula-

tions, we have demonstrated that the MV formalism, within

errors, recovers the bulk flow moments of the underlying

matter distribution and that the MV moments are unbiased

estimators of the bulk flow of a volume of a given scale, re-

gardless of the geometry of a particular survey. The MV mo-

ments are unbiased, in that on average they give the correct

values for the idealized bulk flow components. We calculated

the variance of the bulk estimator using (i) linear theory

?(ui− Ui)2?

Although the variance calculated using the simulations were

found to be somewhat different from the linear theory pre-

dictions, we argued that this is due to the simulations being

underdense and thus not having enough galaxies. For numer-

ical simulations with higher resolution (more galaxies), we

1

2 and (ii) numerical simulations ?(ui− Vi)2?

1

2.

c ? 0000 RAS, MNRAS 000, 000–000

Page 10

10Agarwal & Feldman & Watkins

Figure 8. The window functions W2

calculated using MV (thick) weights (see Eq. 8) and MLE (thin)

weights (see Eq. 3) for R = 20 h−1Mpc (top row) and R =

50 h−1Mpc (bottom row) for the DEEP (left-hand column) and

COMPOSITE (right-hand column) real catalogues. The x,y and

z components are dash-dotted, short-dashed, long-dashed lines,

respectively. The solid line is the ideal window function.

iiof the bulk flow components

expect the Gaussian-weighted moments Vi to approach the

ideal moments Ui and give a much closer match. We found

the variance estimates using simulations and linear theory

to be significantly closer to each other for the DEEP cata-

logue, which has fewer close by galaxies and thus performed

much better than the SFI++ and COMPOSITE catalogues

when testing the MV formalism. These results validate our

use of linear theory in the development of the MV method

and confirms the fact that non-linear, small-scale motions

do not significantly affect the MV estimators.

We tested many facets of the MV formalism and found

agreement in all the tests we performed using the LD and

HR simulations. We found that the chance of getting large

flows (∼ 400 km s−1) in a ΛCDM universe is of order of

∼ 1 per cent. The bulk moments ui estimated using our MV

formalism are, within errors, the same as the moments Vi

of the volume as traced by all the galaxies in the simula-

tion box and linear theory correctly predicts the variance

of the estimators. Further, since the formalism allows for

exploration of all scales where there are data, we can reli-

ably explore flows on many scales and track the dynamics

of volumes of different scales (parametrized by a radius of a

Gaussian sphere R).

6ACKNOWLEDGMENTS

We would like to thank Mike Hudson for many thoughtful

and useful comments. We are also grateful to R´ oman Scocci-

marro and the LasDamas collaboration and Changbom Park

and the Horizon Run collaboration for providing us with the

simulations. This work was supported by the National Sci-

ence Foundation through TeraGrid resources provided by

the NCSA.

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