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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
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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
radialpeculiarvelocitymeasurements are
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
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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)
?
.
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4 Agarwal & 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
3 MOCK CATALOGUES
3.1 N-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.
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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-CarmenHR
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-
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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.
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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.
4BULK 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
8 Agarwal & 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
5 DISCUSSION 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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