Modeling the three-point correlation function
Draft version October 26, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
MODELING THE GALAXY THREE-POINT CORRELATION FUNCTION
Felipe A. Maŕın
, Risa H. Wechsler
, Joshua A. Frieman
and Robert C. Nichol
Draft version October 26, 2018
ABSTRACT
We present new predictions for the galaxy three-point correlation function (3PCF) using high-
resolution dissipationless cosmological simulations of a flat ΛCDM Universe which resolve galaxy-size
halos and subhalos. We create realistic mock galaxy catalogs by assigning luminosities and colors
to dark matter halos and subhalos, and we measure the reduced 3PCF as a function of luminosity
and color in both real and redshift space. As galaxy luminosity and color are varied, we find small
differences in the amplitude and shape dependence of the reduced 3PCF, at a level qualitatively con-
sistent with recent measurements from the SDSS and 2dFGRS. We confirm that discrepancies between
previous 3PCF measurements can be explained in part by differences in binning choices. We explore
the degree to which a simple local bias model can fit the simulated 3PCF. The agreement between
the model predictions and galaxy 3PCF measurements lends further credence to the straightforward
association of galaxies with CDM halos and subhalos.
Subject headings: cosmology: large-scale structure of universe — galaxies: formation — galaxies:
statistics — galaxies: halos
1. INTRODUCTION
Observations of the higher-order statistics of the galaxy
distribution can provide fundamental tests of the stan-
dard cosmological model. For example, higher-order cor-
relation functions of the mass are predicted to be zero
in linear perturbation theory for Gaussian initial condi-
tions, which are expected in the simplest inflation models
of the early Universe (Peebles 1980, Bernardeau et al.
2002, Szapudi 2005 and references therein). In the
late Universe, however, non-linear gravitational cluster-
ing and biased galaxy formation lead to non-Gaussianity
in the galaxy density field, resulting in non-zero con-
nected N-point correlation functions (NPCFs) with N >
2. By studying higher-order galaxy statistics on large
scales, we can test the nature of the initial conditions;
on smaller scales, the NPCFs can constrain models of
biased galaxy formation (e.g., Fry & Gaztañaga 1993;
Frieman & Gaztañaga 1994) and the relationship be-
tween galaxies and their host dark matter halos.
The three-point correlation function (3PCF, or ζ),
and its Fourier-space equivalent, the bispectrum, are
the first in the hierarchy of higher-order statistics and
measure the shape dependence of the number of galaxy
triplets as a function of scale. The 3PCF is sensitive
to, for example, the shapes of dark matter halos and
the presence of filamentary structures in the large-scale
structure of the Universe (Sefusatti & Scoccimarro
2005). Since the pioneering work of Peebles and col-
laborators (see Peebles 1980 and references therein,
1 Department of Astronomy & Astrophysics, Kavli Institute
for Cosmological Physics, The University of Chicago, Chicago, IL
60637 USA
2 Kavli Institute for Particle Astrophysics & Cosmology, Physics
Department, and Stanford Linear Accelerator Center, Stanford
University, Stanford, CA 94305
3 Center for Particle Astrophysics, Fermi National Accelerator
Laboratory, P.O. Box 500, Batavia, IL 60510 USA
4 Institute of Cosmology & Gravitation, University of
Portsmouth, Portsmouth, PO1 2EG, UK
5 e-mail: fmarinp@uchicago.edu
in particular Groth & Peebles 1977), there have been
many measurements of the 3PCF and bispectrum
using a variety of angular (Peebles & Groth 1975;
Groth & Peebles 1977; Gaztañaga & Frieman 1994;
Frieman & Gaztañaga 1999; Huterer et al. 2001;
Szapudi et al. 2001, 2002; Ross et al. 2006) and red-
shift (Gaztañaga & Frieman 1994; Jing & Börner
1998; Verde et al. 1998; Scoccimarro et al. 2001a;
Feldman et al. 2001; Jing & Börner 2004) catalogs.
There has also been considerable theoretical work
to understand the 3PCF and bispectrum using
non-linear perturbation theory (see Bernardeau et al.
2002 and references therein), the halo model (see
Ma & Fry 2000; Scoccimarro et al. 2001b; Wang et al.
2004; Fosalba et al. 2005), and cosmological simulations
(Barriga & Gaztañaga 2002; Scoccimarro et al. 1999;
Gaztañaga & Scoccimarro 2005; Hou et al. 2005).
In recent years, there has been renewed inter-
est in the 3PCF due to the availability of large
redshift surveys. These surveys now provide both
the volume and the number of galaxies required
to make robust measurements of the 3PCF over a
range of scales. For example, recent papers by
Kayo et al. (2004), Hikage et al. (2005), Nichol et al.
(2006), Nishimichi et al. (2006), Kulkarni et al. (2007)
have presented measurements of the 3PCF from the
Sloan Digital Sky Survey (SDSS; York et al. 2000)
as a function of scale, galaxy luminosity, and color.
Nichol et al. (2006) also quantified the effect of large-
scale structures on the shape dependence of the 3PCF.
Likewise, several recent papers (Jing & Börner 2004;
Baugh et al. 2004; Croton et al. 2004; Gaztañaga et al.
2005; Pan & Szapudi 2005; Croton et al. 2006) provide
new measurements of the 3PCF and high-order correla-
tions from the Two-degree Field Galaxy Redshift Survey
(2dFGRS; Colless et al. 2001).
The main results from these recent SDSS and 2dF-
GRS analyses of the 3PCF are: i) on large scales, the
observed 3PCF is in qualitative agreement with expec-
http://arxiv.org/abs/0704.0255v2
2 Maŕın, Wechsler, Frieman and Nichol
tations for the growth of structure from Gaussian initial
conditions; ii) on smaller scales (i.e., in the non-linear
and weakly non-linear regimes), the 3PCF measured in
redshift space scales with the redshift-space two-point
correlation function (2PCF, or ξ) as ζ ∼ ξ2, consis-
tent with the “hierarchical clustering” ansatz (Peebles
1980); iii) the shape dependence of the reduced 3PCF
depends at most weakly on galaxy luminosity; iv) the
amplitude of the 3PCF is larger for elongated triangle
configurations than for more symmetric triangle shapes
(Gaztañaga & Scoccimarro 2005; Gaztañaga et al. 2005;
Nichol et al. 2006; Kulkarni et al. 2007) — again consis-
tent with expectations from non-linear clustering theory.
It is important to make detailed comparisons of these
3PCF observations with theoretical predictions that in-
corporate a realistic prescription for modeling galax-
ies and that account for observational effects such as
redshift-space distortions. Such comparisons can be car-
ried out in two ways: either the observations can be
corrected for the redshift-space effects and compared to
theory in real space, e.g., using the projected 3PCF
(Zheng 2004), which is analogous to the projected 2PCF
(Zehavi et al. 2005), or one can build mock galaxy cata-
logs from cosmological simulations and measure the the-
oretical 3PCF directly in redshift space. Since we also
have access to the real-space 3PCF from such mock cata-
logs, we can investigate in detail the relationship between
the real- and redshift-space correlation functions.
In this paper, we pursue the second of these method-
ologies, using state-of-the-art high-resolution dissipation-
less dark matter cosmological simulations. These simu-
lations have the spatial resolution required to identify
the dark matter (DM) halos and subhalos that host indi-
vidual galaxies and at the same time encompass a large
enough volume to probe large-scale structure in a sta-
tistically reliable way. The model we use assigns galaxy
properties (luminosity, color, etc.) directly to these DM
galactic halos and subhalos using simple, empirically-
based assumptions about these properties. This ap-
proach differs in both assumptions and the resolution
required from methods which build galaxy catalogs by
statistically assigning several galaxies to each (more mas-
sive) halo using a Halo Occupation Distribution (HOD;
e.g. Berlind & Weinberg 2002; Wang et al. 2004; see
Kulkarni et al. (2007) for constraints on HOD param-
eters from the SDSS Luminous Red Galaxy sample’s
3PCF) or from semi-analytic methods. The method used
here was first implemented by Kravtsov et al. (2004),
and has been applied successfully to different statisti-
cal studies, including the 2PCF (Conroy et al. 2006),
galaxy-galaxy lensing (Tasitsiomi et al. 2004), and close
pair statistics (Berrier et al. 2006) among others (see also
Vale & Ostriker 2004, 2006; Conroy et al. 2007). Here
we extend the study of this model to the 3PCF as a
function of luminosity, color, and redshift. Where pos-
sible, we make direct comparisons in redshift space with
recent observations.
In §2, we describe the simulations used in this paper
and our methods for constructing mock galaxy catalogs
based on resolved DM halos. We also review the tech-
niques used to estimate the NPCFs. In §3, we present
measurements of the 3PCF in both real and redshift
space for both the dark matter and galaxy catalogs, while
in §4, we study the dependence of the model 3PCF on
galaxy luminosity and color. In §5, we compare the
model 3PCF with SDSS observations and discuss the ef-
fects of binning. We relate the 3PCF to a simple non-
linear bias model in §6. In §7, we summarize and discuss
our findings.
2. METHODS
2.1. Dark matter simulations
We investigate clustering statistics using cosmological
N -body simulations of structure formation in the concor-
dance, flat ΛCDM cosmology with ΩΛ = 0.7 = 1 − Ωm,
h = 0.7, and σ8 = 0.9, where Ωm, ΩΛ are the present
matter and vacuum densities in units of the critical den-
sity, h is the Hubble parameter in units of 100 km s−1
Mpc−1, and σ8 specifies the present linear rms mass fluc-
tuation in spheres of radius 8 h−1Mpc. The simulations
used here were run using the Adaptive Refinement Tree
N−body code (ART, see Kravtsov et al. 1997 for de-
tails), which implements successive refinements in space
and time in high-density environments. The primary
simulation box we use is 120 h−1Mpc on a side (hereafter,
L120); the number and mass of each dark matter particle
are Np = 512
3 ≈ 1.34× 108 and mp = 1.07× 10
9h−1M⊙
respectively. This simulation has been previously used
to measure several properties of dark matter halos and
subhalos (e.g., Allgood et al. 2006; Wechsler et al. 2006;
Conroy et al. 2006; Berrier et al. 2006). In order to in-
clude more massive halos and study the effects of the
size of the sample on the statistical analysis, we also use
a second simulation with the same cosmological param-
eters in a bigger box, with 200 h−1Mpc on a side (which
was also used to measure halo shapes in Allgood et al.
2006). This box contains Np = 256
3 particles with mass
mp = 3.98× 10
10 h−1M⊙, therefore it will lack low mass
(and luminosity) objects that are included in the L120
From these dark matter samples, virialized concentra-
tions of particles are identified as halos. In order to find
these halos and their constituent subhalos (concentra-
tions of virialized matter inside bigger halos), a variant
of the Bound Density Maxima halo finding algorithm of
Klypin et al. (1999) is used. This algorithm assigns den-
sities to each particle using a smoothing kernel on the 32
nearest neighbors; centering on the highest-overdensity
particle, each center is surrounded by a sphere of radius
rfind = 50h
−1kpc. The algorithm removes unbound par-
ticles when calculating the properties of the halos. The
halo catalog is complete for halos with more than 50
particles, which corresponds to a minimum halo mass of
1.6×1010 h−1M⊙ for the L120 box and 2.0×10
12 h−1M⊙
for halos in the L200 box.
Henceforth, we will use the terms “distinct halo” to
mean any halo that is not within the virial radius of a
larger halo, “subhalo” to indicate a halo that is within
the virial radius of a larger halo, and “galactic halo” to
refer to the halo directly hosting a galaxy. Using this
terminology, the galactic halo of a satellite galaxy is a
subhalo while the galactic halo of a central or isolated
galaxy will be a distinct halo.
2.2. From halos to ‘galaxies’
It is expected that galaxy properties depend in detail
not only on the dark matter clustering, but also on the
Modeling galaxy three-point statistics 3
gas dynamics, radiative processes, and feedback mecha-
nisms that affect the baryonic components. A program
to include all those physical processes in building mock
galaxy catalogs from simulations would require introduc-
ing a number of free parameters and assumptions which
could partially obscure the relevant mechanisms that de-
termine the shape and amplitude of the 3PCF. Our ap-
proach is instead more empirical—to associate galaxies
of given properties with simulated dark matter halos and
subhalos, using halo properties that we can measure in
the simulation, and to see whether this one-to-one corre-
spondence predicts a galaxy 3PCF that is consistent with
the observations. As described below, the assignment of
galaxies to halos was designed to reproduce certain fea-
tures of the observed galaxy distribution, but the 3PCF
was not one of these. As a result, the 3PCF constitutes
a non-trivial test of this approach.
The primary galaxy samples used here are created by
assigning galaxy luminosities and colors drawn from the
SDSS redshift survey to dark matter halos and subhalos,
using the maximum circular velocity at z = 0, Vmax, as
an indicator of the halo virial mass. Vmax has been found
to be a good proxy of the galaxy potential well, which
is a good indicator of stellar mass (Kravtsov et al. 2004;
Conroy et al. 2006).
A galaxy luminosity in the r−band is assigned to
each halo by matching the cumulative velocity function
n(> Vmax) of all galactic halos (distinct halos and sub-
halos) to the observed SDSS r-band luminosity function
(Blanton et al. 2003b) at z = 0.1 (the approximate mean
redshift of the main spectroscopic SDSS galaxy sample
used to estimate the luminosity function). To correct
to z = 0 magnitudes, we use the code kcorrect v3 2
(Blanton et al. 2003a). Since the limited size of the box
gives us an upper limit on the luminosities which can
be reliably studied (in the statistical sense), and at the
same time we cannot sample the lowest-luminosity ob-
jects due to limited spatial resolution, for the L120 box
we present results for galaxies in the absolute magnitude
range −19 ≥ Mhr ≥ −22, where M
r ≡ Mr − 5 logh.
The L200 box has a lower spatial resolution, therefore
it contains only brighter objects, with Mhr
∼ − 20. In
order to assign colors to the galaxies, we use the proce-
dure described by Wechsler (2004) and Tasitsiomi et al.
(2004). This method uses the relation between local
galaxy density (defined as the distance to the tenth near-
est neighbor brighter than Mhr = −19.7 and within
cz = 1000 km/s) and color observed in the SDSS, using
the CMU-Pitt Value Added Catalog constructed from
DR1 (Abazajian et al. 2003), to assign a color to each
mock galaxy. We use the distant observer approxima-
tion (Bernardeau et al. 2002) to obtain positions in red-
shift space. Table 1 describes the subsamples used in this
study.
Although the general association of galaxies with dark
matter subhalos seems quite robust, the detailed asso-
ciation of galaxy properties with subhalo properties is
less clear. While galaxy luminosity is expected to be
quite tightly connected to velocity, the circular velocity
of a given subhalo decreases with time due to tidal strip-
ping as it interacts with its host halo. Observable galaxy
properties such as luminosity and color will likely be less
affected by this process. This implies that galaxy ob-
TABLE 1
Subsamples at z = 0
Box Subsample Number Density
of objects [(h−1Mpc)−3]
L120 All objects Mhr < −19 25371 1.5× 10
L120 −19 < Mhr < −20 15432 8.9× 10
L120 −20 < Mhr < −21 8153 4.7× 10
L120 red (g − r > 0.7) 10059 5.8× 10−3
L120 blue (g − r < 0.7) 15312 8.9× 10−3
L200 All objects Mhr < −20 43564 5.4× 10
L200 −20 < Mhr < −21 30575 3.8× 10
L200 −21 < Mhr < −22 9921 1.2× 10
L200 red (g − r > 0.7) 17278 2.2× 10−3
L200 blue (g − r < 0.7) 23748 2.9× 10−3
servables may be more strongly correlated with Vmax,acc,
the maximum circular velocity of the halos at the mo-
ment of accretion onto their host, than with the current
maximum circular velocity, Vmax,now. This conclusion
is supported by measurements of two-point statistics on
both large and small scales in simulations (Conroy et al.
2006; Berrier et al. 2006). Motivated by these consider-
ations, we construct galaxy catalogs using both Vmax,acc
and Vmax,now. We also use galaxy catalogs at different
redshifts in order to study the evolution of the 3PCF
with time. All these additional catalogs use the L120 box
and have the same spatial density as the sample of halos
selected by Vmax,now. We note that the model for as-
signing color is also not uniquely determined. It is, how-
ever, sufficient to match the two-point clustering length
for red and blue galaxies and several observed properties
of galaxy clusters (Zehavi et al. 2005). Future measure-
ments of both clustering statistics and of the properties
of galaxies in groups and clusters should help to further
refine these galaxy assignment models.
2.3. Measuring the 3PCF
Just as the 2PCF measures the excess probability of
finding two objects separated by a distance r, the 3PCF
describes the probability of finding three objects in a
particular triangle configuration compared to a random
sample. The probability of finding three objects in three
arbitrary volumes dV1, dV2, and dV3, at positions r1, r2
and r3 respectively, is given by (Peebles 1980)
P = [1 + ξ(r12) + ξ(r23) + ξ(r31) + ζ(r12, r23, r31)]×
n̄3dV1dV2dV3, (1)
where n̄ is the mean density of the objects, rij ≡ ri − rj
the distance between two objects, ξ is the 2PCF, and ζ
is the 3PCF:
ξ(r12)= 〈δ(r1)δ(r2)〉 (2)
ζ(r12, r23, r31)= 〈δ(r1)δ(r2)δ(r3)〉; (3)
here δ is the fractional overdensity in the dark matter
field or in the distribution of galaxies. Since the 3PCF
depends on the configuration of the three distances, it
is sensitive to the 2-D shapes of the spatial structures,
at large and small scales (Sefusatti & Scoccimarro 2005;
Gaztañaga & Scoccimarro 2005). Motivated by the “hi-
erarchical” form of the N-point functions, ζ ∝ ξ2, found
4 Maŕın, Wechsler, Frieman and Nichol
by Peebles & Groth (1975), we use the reduced 3PCF
Q(r, u, α) to present our results:
Q(r, u,α ) =
ζ(r, u, α)
ξ(r12)ξ(r23) + ξ(r23)ξ(r31) + ξ(r31)ξ(r12)
. (4)
This quantity is useful since Q is found to be close to
unity over a large range of scales even though ξ and ζ vary
by orders of magnitude (Peebles 1980). To parametrize
the triangles for the 3PCF measurements, r ≡ r12 sets
the scale size of the triangle, while the shape param-
eters are given by the ratio of two sides of the trian-
gle, u = r23/r12, and the angle between the two sides
of the triangle α = cos−1(r̂12 · r̂23), where r̂12, r̂23
are the unit vectors of the first two sides. Following
Gaztañaga & Scoccimarro (2005), triangles where α is
close to 0◦ or 180◦ are referred to as “elongated config-
urations”, while those with α ∼ 50◦ − 120◦ are referred
to as “rectangular configurations”,
We calculate the 2PCF using the estimator of
Landy & Szalay (1993),
DD − 2DR+RR
. (5)
Here, DD is the number of data pairs normalized by
ND ×ND/2, DR is the number of pairs using data and
random catalogs normalized by NDNR, and RR is the
number of random data pairs normalized by NR×NR/2.
The 3PCF is calculated using the Szapudi & Szalay
(1998) estimator:
DDD − 3DDR+ 3DRR−RRR
, (6)
where DDD, the number of data triplets, is normalized
by N3D/6, and RRR, the random data triplets, is nor-
malized by N3R/6. DDR is normalized by N
DNR/2, and
DRR by NDN
We estimate the errors using jack-knife re-sampling.
From each galaxy catalog, we construct sixteen subsam-
ples of L120 or L200; within each of them we remove a
different region (30 × 602 (h−1Mpc)3 for the L120 box,
and 50×1002 (h−1Mpc)3 for the L200 box). The variance
σJK of Q is calculated as:
σ2JK =
N − 1
(Qi − Q̄i)
2, (7)
where N = 16 is the number of subsamples, Qi is the
value for the i−th subsample, and Q̄i is the mean of the
Qi. We note in passing that the validity of jack-knife
resampling as a method to estimate the errors has not
been explicitly tested with mock catalogs for three-point
statistics. Although it is beyond the scope of this paper,
this may be an interesting topic of future investigation
especially once the statistical power of the measurements
improves.
To compute the 2PCF and 3PCF, we use the NPT
software developed in collaboration with the Auton Lab
at Carnegie Mellon University. NPT is a fast implemen-
tation of the NPCFs using multi-resolution kd-trees to
compute the number of pairs and triplets in a dataset.
For more details and information on the algorithm, see
Moore et al. (2001), Gray et al. (2004), and Nichol et al.
(2006).
3. THE 3PCF OF GALAXIES AND DARK MATTER
We estimate the reduced 3PCF for the distribution of
dark matter and for galaxies for different triangle con-
figurations, focusing on the scale and shape dependence
of the 3PCF. We also investigate its time evolution and
how it depends on the selection criterion for subhalos.
We study Q(r, u, α) in both real and redshift space, in
order to compare our results with current observations
and disentangle galaxy biasing effects from those which
are consequences of redshift distortions.
In order to distinguish scale and, most importantly,
shape effects, and to keep the errors as small as possi-
ble, we have chosen an intermediate-resolution binning
scheme. For studies of equilateral triangles (u = 1 and
α = π/3 rad), we use bins of size ∆ log(r) = 0.1. For
measurements of the shape dependence of the 3PCF, we
use triangles with four different scales r = 0.75, 1.5, 3, 6,
and 9 h−1Mpc, using the shape parameters u = 2, and 15
angular bins separated by ∆α = π/15 rad; the resolution
of the bins is given by ∆rij = ±0.03rij. This resolution is
sufficient to see the most important features of the 3PCF
even on small scales, although it is not sufficient to dis-
tinguish the “finger-of-god” effect at the smallest scales
in redshift space, where Q(α) varies very little except
at very small or elongated angles, where it increases to
many times the mean value (Gaztañaga & Scoccimarro
2005).
3.1. The 3PCF in real space
Figures 1 and 2 show the 3PCF for dark matter par-
ticles and for galaxies from the L120 and L200 simula-
tions. Here we plot results for dark matter (thick solid
line), galaxies in halos selected by Vmax,now (thin solid
line), and galaxies in halos selected by Vmax,acc (long-
dashed line) for z = 0 and for halos selected by Vmax,now
at z = 1 (short-dashed line) and z = 2 (dotted line).
Figure 1 shows the reduced 3PCF for equilateral trian-
gles, Qeq(r), in real (left panel) and redshift space (right
panel). In real space, the reduced 3PCF for the dark
matter is only weakly scale dependent on small scales,
decreases rapidly with increasing scale around r ∼ 3h−1
Mpc, and falls off more slowly on larger scales. This
behavior is broadly consistent with previous N-body re-
sults (e.g. Scoccimarro et al. 1998) and with expectations
from leading order non-linear perturbation theory on the
largest scales (shown as the thick red long-dashed curve
in Figure 1 left panel), with loop-corrected perturbation
theory on intermediate scales where the rms perturba-
tion amplitude δ(r) is of order unity (the transition to
the strongly non-linear regime; adding more orders to the
calculation would increase the agreement to the N -body
dark matter 3PCF amplitude), and with quasi-stable hi-
erarchical clustering on the smallest scales. Tests with
the L200 box indicate that the downturn in Qeq for dark
matter at scales larger than r ∼ 8h−1 Mpc is likely due
to finite volume effects.
At scales below r ∼ 10h−1 Mpc, the dark matter Qeq is
larger than that for the galaxies; this behavior is broadly
expected if galaxies are more strongly clustered than
(positively biased with respect to) the mass, cf. eqn.(10).
Modeling galaxy three-point statistics 5
Fig. 1.— The reduced 3PCF, Qeq(r), for equilateral triangles in the L120 box. Left: Results in real space. Thick solid line (black): dark
matter; thin solid line (black): galactic halos selected by Vmax,now at z = 0; short-dashed (green): galactic halos selected by Vmax,now at
z = 1; dotted (red): galactic halos selected by Vmax,now at z = 2; long-dashed (blue): galactic halos selected by Vmax,acc at z = 0; thick
long-dashed (red): leading-order perturbation theory, dark matter. Right: Results in redshift space. Line types correspond to the same
dark matter and halo samples as in the left-hand plot. Error bars are calculated using jack-knife resampling and are only shown for one of
the samples for clarity.
At higher redshift, evolution is seen in Qeq(r) that is con-
sistent with expectations from non-linear gravitational
evolution: on the largest scales, the amplitude of Qeq is
unchanging, as predicted from leading order perturbation
theory, while the sharp break associated with the tran-
sition to the strongly non-linear regime moves to larger
scales as the density perturbation amplitude increases
with time.
Comparing results for subhalos selected by Vmax,now
and by Vmax,acc, differences in the amplitude of Qeq ap-
pear on small scales, r . 3 h−1 Mpc. In halo-model
language, on these scales the 3PCF is sensitive to the
internal structure of halos, i.e., to the one- and two-halo
terms, while the three-halo term dominates the 3PCF on
larger scales (Wang et al. 2004; Takada & Jain 2003).
In Figure 2, the top panels show how the reduced 3PCF
depends on triangle shape in real space. In general,
the 3PCF for elongated configurations is greater than
for rectangular configurations. This is a consequence of
the fact that, in non-linear gravitational instability, ve-
locity flows tend to occur along gradients of the density
field (Bernardeau et al. 2002). The 3PCF is larger for
dark matter than for galaxies for all shapes and scales,
although the difference is larger for elongated configura-
tions. The difference in 3PCF amplitude between rect-
angular and elongated configurations is larger on large
scales, in broad agreement with leading-order theoreti-
cal predictions (Bernardeau et al. 2002): on large scales,
the strong shape dependence is determined by pertur-
bative non-linear dynamics; on smaller scales, the shape
dependence is washed out since the coherence between
the velocity and density fields gives way to virialized mo-
tions. This scale dependence of the 3PCF shape is also
reflected in the redshift evolution: in the L120 box (left
panel), the galaxy 3PCFs at z = 1 and 2 (green-dashed
and red-dotted curves) essentially retain the primordial
shape dependence of leading-order non-linear perturba-
tion theory, i.e., at those redshifts, these scales are still
close to the quasi-linear regime. At r = 3 h−1Mpc, the
largest evolution in Q(α) is found for elongated configu-
rations.
As was seen in Figure 1, the effect of changing halo se-
lection from Vmax,acc to Vmax,now on the 3PCF shape ap-
pears only on small scales, r . 1.5h−1 Mpc, i.e., roughly
within the scale of a typical cluster-mass host halo.
On the larger scales probed in the L200 box (right
panel of Figure 2), the galaxy reduced 3PCF (open cir-
cles) tracks the shape of the dark matter 3PCF fairly
well. The difference between the galaxy and dark matter
3PCF amplitudes on these scales is reasonably well fit
by a simple bias prescription: the thin blue curve is the
biased 3PCF that results from fitting the galaxy 3PCF
with eqn. (10); see §6. We also see that the jack-knife
errors increase on the largest scales, where the effects of
the finite box size start to become evident. For compar-
ison, the red long-dashed curve is the 3PCF of the dark
matter from leading-order non-linear perturbation the-
ory (Bernardeau et al. 2002; Jing & Borner 1997). On
the largest scales, it is in reasonable agreement with the
measured 3PCF for the dark matter.
3.2. The 3PCF in redshift space
Redshift distortions have been studied in depth (and
are a useful tool to constrain cosmological parameters)
for the power spectrum (e.g., Bernardeau et al. 2002;
da Ângela et al. 2005; Tinker 2007) and for the bis-
pectrum (Scoccimarro et al. 1999; Verde et al. 2002;
Sefusatti et al. 2006). Some comparisons have been
made for the 3PCF as well (Matsubara & Suto
1994; Takada & Jain 2003; Wang et al. 2004).
Gaztañaga & Scoccimarro (2005) found that the
redshift distortions do not have a strong dependence on
the cosmological parameters.
The right panel of Figure 1 and the bottom panels
6 Maŕın, Wechsler, Frieman and Nichol
0 50 100 150
0 50 100 150 0 50 100 150
0 50 100 150
0 50 100 150
Fig. 2.— Measurement of the reduced 3PCF as a function of triangle shape, Q(α), for different scales r, with side ratio u = r23/r12 = 2
fixed. Left: Results in the L120 box for r = 0.75, 1.5, and 3 h−1Mpc (from left to right) in real space (top panels) and redshift space
(bottom panels) for dark matter and galaxies; line types correspond to the same dark matter and halo samples as in Figure 1. Right:
Results in the L200 box for r = 6 (left) and 9 h−1Mpc (right) in real (top) and redshift space (bottom); Thick solid line (black): dark
matter; open circles: galaxies in halos selected by Vmax,now at z = 0; thin solid line (blue): predicted galaxy 3PCF using the dark matter
3PCF Qdm and eqn. (10) with best-fit bias parameters c1, c2 obtained from fitting the galaxy 3PCF at r = 9h
−1 Mpc (see §6); long
dashed (red): leading order non-linear perturbation theory prediction for dark matter reduced 3PCF. Error bars calculated using jack-knife
resampling method.
of Figure 2 show the 3PCF in redshift space. The first
feature that can be seen is a dramatic decrease in the
amplitude and in the scale and shape dependence of Q
compared to the real-space measurements. For exam-
ple, for equilateral triangles, the redshift space Qz(r)
is reduced compared to the real space Q(r) at small
scales and increased with respect to the real-space re-
sults on larger scales. The overall effect is that Qz(r)
is nearly scale-independent, i.e., the clustering appears
more hierarchical in redshift space (Suto & Matsubara
1994; Matsubara & Suto 1994; Scoccimarro et al. 1999).
Moreover, in redshift space, the suppression of the galaxy
3PCF relative to that of the dark matter is apparent
on all scales; it appears to be relatively independent of
scale and configuration and is larger than the relative
suppression in real space, consistent with earlier results
(Wang et al. 2004; Gaztañaga et al. 2005). It also ap-
pears that there is very little redshift evolution of the
galaxy 3PCF in redshift space; the measurements at
z = 1 and z = 2 are nearly indistinguishable from each
other. With regard to halo selection, as in real space we
find that QVmax,now < QVmax,acc, but the differences
between them are smaller than in real space.
Together, these results suggest that the shape and scale
dependence of the reduced 3PCF in redshift space on the
scales shown here are largely determined by redshift dis-
tortions, with non-linear gravitational evolution playing
a subdominant role.
4. OBSERVING THE 3PCF: LUMINOSITY AND COLOR
DEPENDENCE
To investigate the dependence of the three-point clus-
tering on galaxy luminosity and color and to make direct
comparisons with measurements from recent redshift sur-
veys, we calculate the 3PCF for galaxies with luminos-
ity and color cuts similar to those that have been ap-
plied to redshift survey data samples, with luminosity
and color information obtained as described in §2. The
luminosities are assigned according to Vmax,now in the
L120 and L200 boxes, since we hace those measurements
for both boxes. As seen in the previous section, the use
of Vmax,now is justified since the differences with respect
to Vmax,acc are only significant on small scales (r<∼ 1.5
h−1Mpc) in real space and are almost completely dis-
sapear on redshift space. We adopt the same binning
scheme used in the previous section.
4.1. Luminosity dependence
The left panels of Figure 3 show results for the re-
duced 3PCF for equilateral configuratons in two lumi-
nosity bins, in real and redshift space. For equilateral
triangles, there is a small difference in 3PCF between
the luminosity samples in real space: the fainter galaxies
have larger Q(r) than the brighter ones, as expected in
a simple linear bias model (eqn. 10). The redshift-space
3PCFs for these galaxies are roughly constant with r,
Qz,gal(r) ∼ 0.7, in agreement with SDSS measurements
for these configurations (see Figures 7-9 in Kayo et al.
2004). There is a very slight difference between the re-
duced 3PCF amplitudes for different luminosity bins in
redshift space, but it is not statistically significant for a
dataset of this size. This result qualitatively agrees with
the SDSS results of Kayo et al. (2004) who also found
almost no luminosity dependence of the reduced 3PCF
on these scales. They found a slightly higher amplitude
for the reduced 3PCF for the −19 > Mhr > −20 sample
compared to that of brighter galaxies, but their results
for the two luminosity samples were consistent within
the error bars. The brightest galaxies (dotted line) show
significant fluctuations in both real and redshift space;
we think this behavior is due to the small size of the box
Modeling galaxy three-point statistics 7
by luminosity
real space
by luminosity
redshift space
by color
real space
by color
redshift space
Fig. 3.— The reduced 3PCF, Q(r), for equilateral triangles as a
function of galaxy luminosity and color in real and redshift space.
Top left : Qeq(r) in real space for galaxies divided into luminosity
bins; long dash-dotted (cyan): −19 > Mhr > −20; short-dashed
(green): −20 > Mhr > −21; dotted (magenta): −21 > M
r > −22.
The brightest sample comes from the L200 box, the other two are
from the L120 box. Bottom left: Results in redshift space; line
types correspond to the same galaxy samples as in the top left
panel. Top right: Qeq(r) in real space for galaxies divided accord-
ing to color, using the L120 box; long-dashed (red): red galaxies
(g− r > 0.7); short dash-dotted (blue): blue galaxies (g− r < 0.7).
Bottom right Results in redshift space; line types correspond to
the same galaxy samples as in the top right panel. Error bars are
calculated using jack-knife respampling.
and the low density of objects.
Figure 4 shows the dependence of Q(α) on galaxy lu-
minosity in real (curves) and redshift space (points). We
use the same ordinate scale for all the plots to empha-
size where configuration effects are more important. The
top panels show results for small scales, calculated with
the L120 box. On these scales, the reduced redshift-
space 3PCF of the brighter sample (−20 > Mhr > −21,
short dashed curve for real space, filled triangles for red-
shift space) is slightly lower than for the fainter sample
(−19 > Mhr > −20, long dashed-dotted curve for real
space, filled squares for redshift space) for all angles,
consistent with the results for Qz(r) using equilateral
triangles. The lower plots in Figure 4 show the lumi-
nosity dependence on larger scales, measured using the
L200 box; note that the luminosity bins in the lower
plots are −20 > Mhr > −21 and −21 > M
r > −22.
The characteristic U-shape of Q(α) appears clearly in
the redshift-space measurements on scales larger than
r = 6 h−1Mpc. The luminosity dependence of Q on
these scales appears non-existent in redshift space and
only slight in real space. There are hints that the reduced
3PCF for fainter galaxies may have slightly higher am-
plitude and stronger shape dependence than for brighter
galaxies, but these trends are not statistically significant
in the samples studied here. Rather, the strong lumi-
nosity dependence observed for the 2PCF (Zehavi et al.
2005) appears to be closely matched by a correponding
dependence of the 3PCF, such that the reduced 3PCF Q
is roughly independent of luminosity.
L120 box L120 box
0 50 100 150
L200 box
0 50 100 150
L200 box
Fig. 4.— Q(α) shape dependence for galaxies divided by lumi-
nosity, for r = 1.5, 3, 6, and 9 h−1Mpc, with fixed u = 2 in
real (lines) and redshift space (symbols). Top two plots are re-
sults for the L120 box: filled squares and long dash-dotted (cyan):
−19 > Mhr > −20; filled triangles and short-dashed (green):
−20 > Mhr > −21; triangles are slightly shifted to the right for
clarity. Bottom plots show results for the L200 box: filled trian-
gles and short-dashed (green): −20 > Mhr > −21; filled circles and
dotted (magenta): −21 > Mhr > −22; circles are slightly shifted
to the right for clarity. Error bars are calculated using jack-knife
resampling and are shown only for one of the samples.
L120 box L120 box
0 50 100 150
L200 box
0 50 100 150
L200 box
Fig. 5.— Q(α) shape dependence for galaxies divided by color,
in the L120 box (top) and L200 box (bottom) in real (lines) and
redshift space (symbols). Open squares and long-dashed line (red):
red (g − r > 0.7) galaxies. open triangles and short dash-dotted
(blue): blue (g − r < 0.7) galaxies; squares are slightly shifted
to the right for clarity. Error bars are calculated using jack-knife
resampling and are shown only for one of the samples.
4.2. Color dependence
The right panels of Figure 3 show the reduced 3PCF
Q(r) for equilateral triangles separately for red and blue
galaxies in real (top) and redshift space (bottom). Due
to limited statistics, we use the full luminosity range in
each color bin, as opposed to Kayo et al. (2004), who di-
8 Maŕın, Wechsler, Frieman and Nichol
vided the color bins into luminosity subsamples as well
(see their Figure 9). Since the luminosity functions for
red and blue galaxies differ, our red and blue samples
have different characteristic luminosities — the red sam-
ple is on average brighter. In both real and redshift space,
the red galaxies appear to have a slightly higher reduced
3PCF amplitude. Comparing with the left panels, this
difference is in the opposite sense from that expected
from the fact that the red galaxies are brighter; put an-
other way, if we were to compare red and blue samples
of the same luminosity, the color difference in Q would
likely be larger than that seen here. On the other hand,
we should not overinterpret these trends, since the differ-
ences are within the statistical errors. Kayo et al. (2004)
found a similarly weak dependence of Q in redshift space
on color.
Figure 5 shows the color dependence of Q(α) in real
(curves) and redshift space (points), for the same sam-
ples and configurations as in Figure 4. In real space, on
scales larger than about 3 h−1 Mpc, Q for red galaxies
is larger for rectangular configurations and smaller for
elongated configurations than for blue galaxies. This be-
havior is qualitatively consistent with a picture in which
red galaxies preferentially occupy the inner regions of
clusters, while blue galaxies tend to trace out more el-
liptical or filamentary structures. In redshift space, the
trend with color is largely washed out. These redshift
space results appear more consistent with the observa-
tions of the 2dFGRS (Gaztañaga et al. 2005), where the
color differences for Q in redshift space are smaller than
those seen in the SDSS by Kayo et al. (2004).
The fact that red galaxies have a larger two-point clus-
tering amplitude than blue galaxies and that the reduced
3PCF for red galaxies is also larger than for blue galax-
ies (at least for rectangular configurations) suggests that
red galaxies have a larger quadratic (non-linear) bias,
Cf. eqn. 10. This is qualitatively consistent with the ob-
served morphology-density or color-density relation ac-
cording to which red galaxies are preferentially found in
dense regions, since the latter contribute more strongly
to the higher-order correlations. The qualitative agree-
ment between the two- and three-point observations and
our method for assigning galaxy colors confirms that the
color of a galaxy, a consequence of many physical pro-
cesses occurring inside the galaxies, depends largely on
the surrounding environment. However, we note that
the color assignment for these subhalos is the most un-
certain part of the model; future work will be required
to determine how clustering statistics will change if more
sophisticated schemes are adopted.
4.3. Redshift distortions: Galaxy Type and Evolution
So far, we have explored some differences between mea-
surements in real and redshift space of the 3PCF for dif-
ferent simulated galaxy samples. Here we investigate in
more detail whether the redshift distortions of the 3PCF
are universal or instead depend on the type of galaxy
studied and to what extent they evolve with time. We
study the behavior of the quantity
∆Q(r, u, α) ≡ Qz(r, u, α)−Qr(r, u, α), (8)
where Qr, Qz represent the reduced 3PCF measured in
real and redshift space, respectively. We explore ∆Q
L120 box
dark matter
L120 box
blue galaxies
red galaxies
L120 box
L200 box
Fig. 6.— ∆Q(r, u, α) ≡ Qz(r, u, α)−Qr(r, u, α) as a function of
galaxy type and redshift. Top: ∆Q for dark matter at z = 0 and for
galaxy samples at different redshifts in the L120 box, for equilateral
triangles (top left) and as a function of angle for triangles with
r = 3 h−1Mpc and u = 2 (top right). Bottom: ∆Q as a function
of galaxy type for equilateral triangles in the L120 box (bottom
left) and as a function of angle for r = 6 h−1Mpc and u = 2 in the
L200 box (bottom right).
as a function of galaxy type (luminosity and color) and
epoch. Figure 6 shows ∆Q for equilateral triangles as a
function of scale and also the configuration dependence
for triangles with fixed r, u = 2, and different opening
angles α.
In general, the trends are similar to those seen above:
Qz < Qr for small scales and for elongated triangle con-
figurations, while the opposite behavior is seen for larger
scales and for rectangular configurations. At z = 0, for
equilateral triangles ∆Q(r) appears to display a roughly
universal scale dependence, independent of galaxy type,
with ∆Q(r) ≃ 0.67r−2 − 2.6r−1 − 0.02r + 0.88 over the
range 0.5 ≤ r ≤ 5 h−1Mpc. On the other hand, for
r = 6h−1 Mpc and u = 2, the shape dependence of
∆Q shows more dependence on galaxy type, with blue
galaxies having larger values than red galaxies and bright
galaxies smaller values than faint galaxies. Note that the
redshift distortions of Q appear insensitive to whether
the subhalos are identified at the present or at the time
they are first accreted onto host halos.
We see clear evolution of ∆Q with redshift, as expected
since redshift distortion effects should become more pro-
nounced as perturbations become more non-linear. For
equilateral triangles, the upper left panel of Figure 6
shows that the scale r where ∆Q ∼ 0 increases with
time. The shape dependence of ∆Q at fixed scale also
appears to increase with time.
5. OBSERVING THE 3PCF: COMPARISON TO SDSS DATA
AND BINNING EFFECTS
Below, we compare our results with measurements of
the 3PCF from the SDSS by Nichol et al. (2006). The
SDSS sample is magnitude limited, with mr < 17.5, and
has additional cuts in absolute magnitude, −19 < Mr <
−22, and in redshift, 0.05 < z < 0.15. In order to com-
pare with this data, we randomly resample the galaxies
from the L120 simulation (in particular, the Vmax,noew
Modeling galaxy three-point statistics 9
Fig. 7.— Top: Qz(r, u = 2, α) at r = 1.0, 2.0 and 4.0 h
−1Mpc
from SDSS observations (green points) from Nichol et al. (2006)
and from our pseudo-flux-limited-sample in the L120 box (long-
dashed lines) with the same binning scheme used in the mentioned
paper. Bottom: Effect of binning in the 3PCF measurements.
Qz(r, u = 2, α) at r = 2.0 and 4.0 h
−1Mpc in the L120 box and
r = 10 h−1Mpc in the L200 box for the simulations in a volume-
limited sample. solid (red): results using a wide binning scheme
(Nichol et al. 2006); long-dashed (black): results with narrow bin-
ning. In the middle panel, we plot the results using SDSS data
with wide (green points) and narrow (triangles) binning.
sample at z = 0), so that they have the same absolute
magnitude distribution as the SDSS flux-limited sample
used in Nichol et al. (2006); we call these pseudo-flux-
limited samples. This resampling technique does not
properly model the distance-dependent selection func-
tion of the SDSS, but it should reproduce its clustering
properties on average. Moreover, since, as we have just
shown, the luminosity dependence of the reduced 3PCF
is weak, we expect that this procedure should be suffi-
ciently accurate for our purposes.
Nichol et al. (2006) measure the shape dependence of
Q(α) for four different length scales, using wide bins in
r, u, and α (see their paper for more details): ∆r = 1
h−1Mpc, ∆u = 1 and ∆α = 0.1 rad. In Figure 7,
we compare the SDSS results (green points) to the cal-
culation of Qz(r, u = 2, α) using our model redshift-
space, volume-limited sample 3PCF (solid lines) and
our pseudo-magnitude-limited sample (dotted line), each
with the same binning as the data. We use the L120 box
for small scales and the L200 box for the r = 10 h−1Mpc
measurements. As the top panels of Figure 7 show, the
model and data show good agreement within the model
jack-knife error bars in amplitude as well as in the shape
of Q(α).
Over the range of scales shown here, both the SDSS
data and the simulations show relatively little shape de-
pendence for the reduced 3PCF amplitude. This is in
contrast to the results of §4 above and to the 2dFGRS re-
sults of Gaztañaga et al. (2005) on similar scales, where
a significantly stronger shape dependence of Q(α) is evi-
dent. As noted by Gaztañaga & Scoccimarro (2005) and
Kulkarni et al. (2007), the differences can be traced to
the binning scheme: the relatively wide binning scheme
used here results in smearing and therefore suppression
of the U-shape of Q over most scales of interest. This
effect is not due primarily to the binning in α: small
bins in both r and u are necessary to see the effects of
shape-dependent clustering.
To illustrate these effects in more detail, in the lower
panels of Figure 7 we also show results for the same sam-
ple but with a narrower binning scheme, using ∆r =
0.1 h−1Mpc (ten times smaller), ∆u = 0.2 (five times
smaller), and ∆α = 0.05 rad (two times smaller). With
the narrower bins (shown by the dashed curves in Figure
7), the shape-dependence ofQ is more pronounced for the
simulation, especially on larger scales. For comparison,
for r = 4 h−1Mpc (lower center panel), the solid trian-
gles shows results for the SDSS flux-limited sample us-
ing a similar narrow binning scheme Nichol et al. (2006),
again showing good agreement between the model and
the data.
6. GALAXY BIAS AND THE 3PCF
As we have seen, the 3PCF predicted for galax-
ies differs systematically from that expected for dark
matter. These differences reflect differences in the
spatial distributions of these two populations; higher-
order statistics can therefore provide important con-
straints upon the bias between galaxies and dark mat-
ter (Fry & Gaztañaga 1993; Frieman & Gaztañaga 1994)
and its dependence upon galaxy properties.
On large scales, where the rms dark matter and galaxy
overdensities are small compared to unity, it is com-
mon to adopt a deterministic, local bias model (e.g.
Fry & Gaztañaga 1993),
δgal = f(δdm) = b1δdm +
δ2dm + ..., (9)
where δgal and δdm are the local galaxy and dark matter
overdensities smoothed over some scale R. We can use
the simulations above to test how well this simple bias
prescription characterizes the galaxy distribution and its
clustering statistics.
Figure 8 shows the relation between δgal and δdm for
all subhalos in the L200 box. The points show the over-
densities for the galaxy and dark matter fields in ran-
domly placed spheres of radius 10h−1 Mpc in both real
(left panel) and redshift space (right panel). The solid
black curves show the best quadratic fits of the form
in eqn. (9). The quadratic local bias model appears
to do a reasonable job in characterizing the mean rela-
tion. Nonetheless, there is significant scatter, either due
to stochastic bias or dependence of bias on other prop-
erties than δdm, that is not captured by this simple bias
model. The errors on these fits, also extended to samples
divided by galaxy luminosity, are shown in Fig. 9 by the
light solid contours.
To leading order, this bias prescription leads to a
relation between the galaxy and dark matter reduced
3PCF amplitudes of the form (Fry & Gaztañaga 1993;
Gaztañaga & Scoccimarro 2005)
Qgal =
(Qdm + c2) , (10)
where c1 = b1 and c2 = b2/b1. Also to leading order,
at low overdensities the relation between the galaxy and
dark matter 2PCF amplitudes in this model is given just
10 Maŕın, Wechsler, Frieman and Nichol
Fig. 8.— Galaxy vs. dark matter overdensities for all galaxies
in the L200 box measured in randomly placed spheres of radius
10h−1 Mpc in real (left panel) and redshift space (right panel).
Solid black curve denotes best fit using the quadratic bias relation
of eqn. 9. Long-dashed red line indicates the best linear bias fit
(b2 = 0) to the 2PCF, i.e., estimating b
= ξgal/ξdm. Short-dashed
blue curve indicates best fit of eqn. 10 to the 3PCF of the galaxies
and dark matter.
by the linear bias, ξgal = b
1ξdm. We can test how well
this bias prescription captures the clustering statistics
by fitting these relations to the dark matter and galaxy
2- and 3-point correlation functions in the simulation in
both real and redshift space and extracting the parame-
ters c1 and c2. Since the relation (9) holds for the den-
sity fields on some smoothing scale, we should fit the
correlation functions for separations comparable to or
slightly larger than this scale. For the 3PCF, we use
triangles with u ≡ r23/r12 = 2, r = 9 h
−1Mpc, and
weight equally all configurations with 00 < α < 1800
using the L200 box. To calculate the likelihood func-
tion, we use a method similar to that described in
Gaztañaga & Scoccimarro (2005), which is based on an
eigenmode analysis of the covariance matrix. We use
the jack-knife subsamples in real and redshift space to
construct covariance matrices, and we use only the dom-
inant eigemodes with values >
2/N where N = 16
is the number of jack-knife subsamples; we found that
adding further eigenmodes just increases artificially the
signal.
The best-fit parameter values from the 3PCF, substi-
tuted into Eqn. 9, are shown as the short-dashed blue
curves in Figure 8. Also, in the right panels of Fig-
ure 2 we show the predicted galaxy 3PCF (solid blue
curves) using the measured dark matter Qdm and the
best fitting c1, c2 parameters in Eqn. (10) from the fit at
r = 9h−1 Mpc. We see that the agreement with the mea-
sured galaxy 3PCF is very good in both real and redshift
space for triangles with r = 9h−1 Mpc; for configurations
with r = 6h−1 Mpc, the agreement in real space is still
quite good while in redshift space some deviations in the
shape-dependence appear.
In Figure 9 we show the fits for c1 and c2 for the dif-
ferent galaxy samples in the L200 box in both real and
redshift space: all galaxies (top panels), galaxies in the
absolute range −20 > Mhr > −21 (middle), and those in
the range −21 > Mhr > −22 (bottom). The thick oval
contours indicate the 1 and 2σ confidence intervals for a
∆χ2 distribution with two free parameters, constrained
using the 3PCF, via Eq. (10), and the large points in-
dicate the maxiumum likelihood values from the 3PCF.
The thin solid contours show the constraints on c1 and c2
using the fit of Eq. (9) to the measurements in Figure 8,
and the vertical line in each panel shows the estimate of
c1 from comparing the 2PCF amplitudes for galaxies and
dark matter using pairs selected from the triplets used to
measure the 3PCF. The best-fit values are shown in Ta-
ble 2: the first two columns show the best-fit parameters
1 , c
2 using the reduced 3PCF (eq. 10), the next two
columns, cδ1 and c
2, are calculated using the quadratic
bias model (eq. 9) fit to the counts in cells, and in the
last column c
1 is obtained from comparing the 2PCF of
galaxies and dark matter assuming a linear bias model,
c21 = ξgal/ξdm.
In agreement with previous measurements in surveys
and simulations, the bias parameters obtained from the
3PCF are degenerate, resulting in elongated contour el-
lipses; this could be mitigated to some extent by using
a larger variety of triangle configurations. In real space
(left panels of Figure 9), we see that the three methods
of extracting the bias parameters are in rough agree-
ment: there is a preference for c1 ∼ 1 and negative
c2 as was found for the 2dFGRS 3PCF measurements
(Gaztañaga et al. 2005). Note that the 3PCF fit tends
to overestimate c1 and to slightly overestimate c2 com-
pared to the other two methods. In redshift space (right
panels), the opposite is true: the 3PCF constraint tends
to underestimate c1. The right panels of Figs. 8 and
9 show that there is a larger discrepancy between the
3PCF fits and the counts-in-cells fit to Eqn. (9) in red-
shift space, suggesting that the relation (10) may not be
a good representation in redshift space.
While this comparison with the deterministic local
bias model is suggestive, to test it more quantitatively
one should use more triangle configurations and larger-
volume catalogs to enable more precise calculation of the
correlation matrices.
7. SUMMARY
We have studied the 3PCF of dark matter and galax-
ies in high-resolution dissipationless cosmological simu-
lations. The galaxy model associates dark matter ha-
los with galaxies by matching the halo velocity function,
including subhalos, with the observed galaxy luminosity
function by abundance, and has been shown previously to
provide an excellent match to observed two-point statis-
tics for galaxies (Conroy et al. 2006). Our primary re-
sults are as follows:
1. The reduced real-space 3PCF for both galaxies
and dark matter has strong dependence on scale
and shape. The shape dependence of the 3PCF
strengthens with increasing scale, in agreement
with previous simulation results for dark matter
and with expectations from non-linear perturba-
tion theory.
2. On small scales, or alternatively with increasing
time, the shape dependence of Q washes out as
Modeling galaxy three-point statistics 11
TABLE 2
Best-fit bias parameters in the L200 box
Subample c
All objects r-space 1.16
+0.21
−0.15
-0.19
+0.19
−0.16
+0.05
−0.12
-0.20
+0.19
−0.22
1.06±0.09
All objects z-space 0.86
+0.11
−0.14
-0.31
+0.15
−0.11
+0.05
−0.07
-0.32
+0.12
−0.18
1.03±0.08
−20 < Mhr < −21 r-space 1.08
+0.28
−0.18
-0.20
+0.28
−0.23
+0.07
−0.16
-0.08
+0.08
−0.19
1.01±0.09
−20 < Mhr < −21 z-space 0.86
+0.21
−0.16
-0.30
+0.20
−0.15
+0.06
−0.12
-0.34
+0.22
−0.16
1.00±0.08
−21 < Mhr < −22 r-space 1.42
+0.48
−0.29
-0.15
+0.33
−0.35
+0.06
−0.14
-0.36
+0.31
−0.05
1.19±0.10
−21 < Mhr < −22 z-space 0.99
+0.15
−0.35
-0.20
+0.21
−0.30
+0.12
−0.05
-0.43
+0.33
−0.07
1.13±0.13
0.5 1 1.5
0.5 1 1.5
Fig. 9.— Constraints on the model bias parameters c1 and c2 in
real (left panels) and redshift space (right panels), measured in the
L200 box for all galaxies (top), galaxies with −20 > Mhr > −21
(middle), and galaxies with −21 > Mhr > −22 (bottom). Thick
ellipses correspond to 1σ and 2σ constraints on parameters using
the reduced 3PCF fit to equation 10; symbols denote the minimum
χ2 value. Thin ellipses come from fitting the smoothed halo and
dark matter overdensities (equation 9) shown in Figure 8. The
vertical lines show the constraint on c1 from comparing the dark
matter and galaxy 2PCF amplitudes and fitting the ratio with a
linear bias model, c2
= ξgal/ξdm.
virial motions within halos replace coherent infall
on larger scales.
3. Redshift-space distortions attenuate the shape and
scale dependence of the reduced 3PCF and weaken
the evolution with redshift.
4. The reduced 3PCF shows only weak dependence on
galaxy luminosity and color; put another way, the
scaling between the 3PCF amplitude and the 2PCF
is predicted to be nearly independent of galaxy
type in this model. The trend of Q with color is
somewhat stronger than with luminosity: the re-
duced 3PCF is slightly enhanced for red galaxies
over blue, especially for elongated configurations.
5. Our model predictions are in excellent agreement
with the shape and scale dependence of the galaxy
3PCF measured in the SDSS when the same bin-
ning scheme is used. Since the results are highly
sensitive to the binning scheme, caution must be
exercised in comparing theory and observations of
the 3PCF. In combination with earlier results, this
comparison indicates that a simple scheme in which
galaxies and dark matter halos and subhalos are
associated in a one-to-one fashion based on maxi-
mum circular velocity can provide a good match to
a wide range of galaxy clustering statistics.
6. The effect on the 3PCF of changing the selection of
subhalos (i.e., of connecting galaxy luminosity to
Vmax,acc instead of Vmax,now) is evident on small
scales (r < 2 h−1Mpc) and for elongated configu-
rations, but is negligible on larger scales. Future
measurements of the 3PCF will help constrain dif-
ferent models for the association of galaxy luminos-
ity and color with subhalo properties.
7. On scales of order 10h−1 Mpc, a local, determinis-
tic bias scheme is in reasonable agreement with the
galaxy and dark matter distributions of the model.
The bias parameters extracted fromQgal are in rea-
sonable agreement with the δgal-δdm relation in real
space, less so in redshift space. Nevertheless, the
redshift-space constraints on the bias parameters
are in agreement with the 2dFGRS measurements
of the 3PCF.
We are indebted to Anatoly Klypin and Brandon All-
good for running and making available the simulations
used in this paper, which were run on the Columbia ma-
chine at NASA Ames and on the Seaborg machine at
NERSC (Project PI: Joel Primack), to Andrey Kravtsov
for running some of the halo catalogs used in this study,
and to Charlie Conroy for providing us with measure-
ments of vmax,acc. We thank Enrique Gaztañaga for
enlightening comments on measuring and interpreting
the 3PCF and comparing results of Q(α) with his es-
timator; and Cameron McBride for discussions on the
measurements of the bias parameters. We additionally
thank Andrey Kravtsov, Issha Kayo and Roman Scoc-
cimarro for several useful discussions. RHW was par-
tially supported by NASA through Hubble Fellowship
grant HST-HF-01168.01-A awarded by the Space Tele-
scope Science Institute, and also recieved support from
the U.S. Department of Energy under contract number
DE-AC02-76SF00515. This work was supported in part
by the Kavli Institute for Cosmological Physics through
the grant NSF PHY-0114422, and by the U.S. Depart-
ment of Energy at Fermilab and at U. Chicago. FAM
thanks the Fulbright Program and CONICYT-Chile for
additional support.
This study also made use of the SDSS DR3 Archive, for
which funding has been provided by the Alfred P. Sloan
12 Maŕın, Wechsler, Frieman and Nichol
Foundation, the Participating Institutions, the National
Aeronautics and Space Administration, the National Sci-
ence Foundation, the U.S. Department of Energy, the
Japanese Monbukagakusho, and the Max Planck Soci-
ety. The SDSS Web site is http://www.sdss.org/. The
SDSS is managed by the Astrophysical Research Consor-
tium (ARC) for the Participating Institutions: the Uni-
versity of Chicago, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, the Johns Hop-
kins University, Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the Max-
Planck-Institute for Astrophysics (MPA), New Mexico
State University, University of Pittsburgh, Princeton
University, the United States Naval Observatory, and
the University of Washington. We also made extensive
use of the NASA Astrophysics Data System and of the
astro-ph preprint archive at arXiv.org.
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