Statistical Determination of Bulk Flow Motions
ABSTRACT 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 (v_p) 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 v_p=270^{+433}km/s (also consistent with v_p=0) for this cluster sample (at z=0.1), which is in agreement with that predicted for a WMAP5-normalised LCDM model (i.e., v_p(LCDM=203km/s). This measurement does not lend support to recent claims of excessive bulk motions (\simeq1000 km/s) which appear in conflict with LCDM, although our large statistical error cannot rule them out. From the measured coherent evolution of v_p, we develop a technique to re-construct the perturbed potential, as well as estimating the unbiased matter density fluctuations and scale--independent bias. Comment: 8 pages, 5 figures
- [Show abstract] [Hide abstract]
ABSTRACT: We develop an optimized technique to extract density--density and velocity--velocity spectra out of observed spectra in redshift space. The measured spectra of the distribution of halos from redshift distorted mock map are binned into 2--dimensional coordinates in Fourier space so as to be decomposed into both spectra using angular projection dependence. With the threshold limit introduced to minimize nonlinear suppression, the decomposed velocity--velocity spectra are reasonably well measured up to scale k=0.07 h/Mpc, and the measured variances using our method are consistent with errors predicted from a Fisher matrix analysis. The detectability is extendable to k\sim 0.1 h/Mpc with more conservative bounds at the cost of weakened constraint.Monthly Notices of the Royal Astronomical Society 03/2010; 407. · 5.52 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: The mounting evidence for anomalously large peculiar velocities in our Universe presents a challenge for the LCDM paradigm. The recent estimates of the large scale bulk flow by Watkins et al. are inconsistent at the nearly 3 sigma level with LCDM predictions. Meanwhile, Lee and Komatsu have recently estimated that the occurrence of high-velocity merging systems such as the Bullet Cluster (1E0657-57) is unlikely at a 6.5-5.8 sigma level, with an estimated probability between 3.3x10^{-11} and 3.6x10^{-9} in LCDM cosmology. We show that these anomalies are alleviated in a broad class of infrared-modifed gravity theories, called brane-induced gravity, in which gravity becomes higher-dimensional at ultra large distances. These theories include additional scalar forces that enhance gravitational attraction and therefore speed up structure formation at late times and on sufficiently large scales. The peculiar velocities are enhanced by 24-34% compared to standard gravity, with the maximal enhancement nearly consistent at the 2 sigma level with bulk flow observations. The occurrence of the Bullet Cluster in these theories is 10^4 times more probable than in LCDM cosmology. Comment: 15 pages, 6 figures. v2: added referencesPhysical review D: Particles and fields 04/2010;
Page 1
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
Page 2
2
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
Page 3
3
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)
Page 4
4
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)
Page 5
5
σ (Mpc h−1)
π (Mpc h−1)
−60−40−200 20 4060
−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)