# Evidence of Galaxy Assembly Bias in SDSS DR7 Galaxy Samples from Count
Statistics

Kuan Wang1-4, Yao-Yuan Mao5, Andrew R. Zentner3,4, Hong Guo6, Johannes U.
Lange7,8, Frank C. van den Bosch9, and Lorena Mezini3,4
1Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
2Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor,
MI 48109, USA
3Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA
15260, USA
4Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC),
University of Pittsburgh, Pittsburgh, PA 15260, USA
5Department of Physics and Astronomy, Rutgers, The State University of New
Jersey, Piscataway, NJ 08854, USA
6Shanghai Astronomical Observatory, CAS, Shanghai 200030, China
7Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA
8Kavli Institute for Particle Astrophysics and Cosmology and Department of
Physics, Stanford University, CA 94305, USA
9Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT
06511, USA
E-mail<EMAIL_ADDRESS>Einstein Fellow; E-mail<EMAIL_ADDRESS>
(11 April 2022)

###### Abstract

We present observational constraints on the galaxy–halo connection, focusing
particularly on galaxy assembly bias, from a novel combination of counts-in-
cylinders statistics, $P(N_{\mathrm{CIC}})$, with the standard measurements of
the projected two-point correlation function,
$w_{\mathrm{p}}(r_{\mathrm{p}})$, and number density, $n_{\mathrm{gal}}$, of
galaxies. We measure $n_{\mathrm{gal}}$, $w_{\mathrm{p}}(r_{\mathrm{p}})$ and
$P(N_{\mathrm{CIC}})$ for volume-limited, luminosity-threshold samples of
galaxies selected from SDSS DR7, and use them to constrain halo occupation
distribution (HOD) models, including a model in which galaxy occupation
depends upon a secondary halo property, namely halo concentration. We detect
significant positive central assembly bias for the $M_{r}<-20.0$ and
$M_{r}<-19.5$ samples. Central galaxies preferentially reside within haloes of
high concentration at fixed mass. Positive central assembly bias is also
favoured in the $M_{r}<-20.5$ and $M_{r}<-19.0$ samples. We find no evidence
of central assembly bias in the $M_{r}<-21.0$ sample. We observe only a
marginal preference for negative satellite assembly bias in the $M_{r}<-20.0$
and $M_{r}<-19.0$ samples, and non-zero satellite assembly bias is not
indicated in other samples. Our findings underscore the necessity of
accounting for galaxy assembly bias when interpreting galaxy survey data, and
demonstrate the potential of count statistics in extracting information from
the spatial distribution of galaxies, which could be applied to both
galaxy–halo connection studies and cosmological analyses.

###### keywords:

cosmology: large-scale structure of Universe – cosmology: observations –
galaxies: evolution – galaxies: formation – galaxies: haloes – galaxies:
statistics

††pubyear: 2022††pagerange: Evidence of Galaxy Assembly Bias in SDSS DR7
Galaxy Samples from Count Statistics–B

##  1 Introduction

Galaxies form in dark matter haloes, and trace the underlying matter field of
the Universe (White & Rees, 1978; Blumenthal et al., 1984). In the ongoing
campaigns to constrain cosmology and understand galaxy evolution, the spatial
distribution of galaxies serves as a rich source of observational information
(e.g., Tröster et al., 2020; Heymans et al., 2021; Porredon et al., 2021). The
comparison of cosmological theory against galaxy survey data is dependent on
the modelling of galaxy formation and evolution in the large-scale structure
(e.g., Desjacques et al., 2018), specifically, the connection between galaxies
and dark matter haloes (see Wechsler & Tinker, 2018, for a review). In
addition to its use in cosmological analyses, the galaxy–halo connection is
interesting in its own right, as dark matter haloes dominate the gravitational
potential wells within which galaxies form and evolve, and halo evolution may
regulate many physical processes in galaxies and clusters.

One of the most frequently adopted simplifying assumptions for modelling the
galaxy–halo connection is that galaxy properties depend solely on halo mass
(Mo et al., 2004; Kauffmann et al., 2004; Blanton et al., 2006). Models based
upon this assumption successfully capture the gross behaviour of the large-
scale distribution of galaxies. The success of the “halo mass-only” assumption
is evident in the halo occupation distribution (HOD; Zheng et al., 2007), a
widely-applied modelling technique which treats the number of central and
satellite galaxies in a halo above a luminosity threshold as simple functions
of the host halo mass. The HOD model is ubiquitous in cosmological analyses
due, in part, to both its simplicity and its flexibility. However, as
increasing amounts of data demand ever greater accuracy in theoretical models
of nonlinear clustering, the halo mass-only approximation must submit to
increased scrutiny.

Simulations have long suggested that halo properties other than halo mass can
impact galaxy properties. For example, the assembly histories of haloes tend
to correlate with the properties of the galaxies that they host, even at a
fixed halo mass. Both hydrodynamical galaxy formation simulations (e.g.,
Artale et al., 2018; Bose et al., 2019; Xu & Zheng, 2020) and semi-analytical
models (e.g., Croton et al., 2007; Contreras et al., 2019; Zehavi et al.,
2018) show that central galaxy luminosity is higher in older halos at a fixed
halo mass. In addition, both gravity-only (e.g., Zentner et al., 2005; Mao et
al., 2015) and hydrodynamical (e.g., Zehavi et al., 2018) simulations show
that the number of satellite galaxies (or subhaloes in the gravity-only case)
is lower in older halos at a fixed halo mass. These effects are sometimes
referred to as “galaxy assembly bias” (Zentner et al., 2014).

In the presence of different strengths of galaxy assembly bias, the same
galaxy clustering signal can, in fact, be interpreted as different halo
clustering signals and hence different cosmologies. As such, analyses of
galaxy clustering signals with halo mass-only models, such as the HOD, could
be systematically biased (e.g., Wu et al., 2008; Zentner et al., 2014;
McCarthy et al., 2019), and it is imperative that we include galaxy assembly
bias effects properly in galaxy–halo connection models that we use to
interpret galaxy survey data. Some of the more recent empirical galaxy–halo
connection models have attempted to include a tunable strength of galaxy
assembly bias (e.g., Lehmann et al., 2017). One notable model is the decorated
halo occupation distribution (dHOD; Hearin et al., 2016), which is based on
the HOD formalism, but enables the number of central and satellite galaxies to
depend on one additional secondary halo property beyond host halo mass, in our
case the halo concentration. The choice of this additional halo property has
also been discussed by several authors, for example, Xu et al. (2021).

Nevertheless, even equipped with models that include the effects of galaxy
assembly bias, it is not straightforward to obtain robust constraints on the
strength of galaxy assembly bias in observational data sets. In recent years,
increasing effort has gone into detecting galaxy assembly bias
observationally. These efforts are somewhat diverse and involve the
application of distinct analysis techniques to a variety of data sets. One
approach is to construct several different observed galaxy samples at
approximately fixed halo mass and compare the large-scale clustering biases
among these samples. Because it is extremely difficult to build observed
samples of galaxies at fixed halo mass (or samples with identical halo mass
distributions), results from these approaches have been rather mixed (e.g.,
Tinker et al., 2011; Lin et al., 2016; Miyatake et al., 2016; More et al.,
2016; Dvornik et al., 2017; Niemiec et al., 2018; Obuljen et al., 2020; Lin et
al., 2022). Another approach is to compare observed and simulated galaxy
statistics directly (i.e., to adopt a forward modelling approach). This
strategy requires some method for associating galaxies with dark matter
haloes, such as a galaxy–halo connection model that includes galaxy assembly
bias, e.g., the dHOD. In this case, both the choice of the model and the
degeneracy between galaxy assembly bias and other model parameters can
significantly weaken conclusions, leading to mixed results as well (e.g.,
Abbas & Sheth, 2006; Cooper et al., 2010; Kauffmann et al., 2013; Wang et al.,
2013; Lehmann et al., 2017; Vakili & Hahn, 2019; Zentner et al., 2019; Salcedo
et al., 2020; Yuan et al., 2021). These studies leave open the debate over the
existence and potential importance of galaxy assembly bias in studies of
galaxy clustering, galaxy evolution, and cosmology.

One way to improve constraints in the forward modelling approach is to include
additional galaxy clustering statistics to help break degeneracies and more
robustly constrain the galaxy–halo connection. In addition to the projected
two-point correlation function, $w_{\mathrm{p}}(r_{\mathrm{p}})$, which is
commonly used in canonical analyses, other summary statistics have been
introduced. These include the weak lensing signal (Cacciato et al., 2013; Lin
et al., 2016; Lange et al., 2019b), redshift space distortions (Reid et al.,
2014; Guo et al., 2015; Lange et al., 2022), the void probability function
(VPF; Walsh & Tinker, 2019), and nearest neighbour distributions (kNN-CDF;
Banerjee & Abel, 2021), each of which provides distinct information. In Wang
et al. (2019, hereafter W19), we compared several summary statistics,
examining the constraining power of different statistic combinations on the
galaxy–halo connection. We forecast that counts-in-cells statistics are
superior to the VPF and the weak lensing signal in complementing
$n_{\mathrm{gal}}$ and $w_{\mathrm{p}}(r_{\mathrm{p}})$. We showed that the
combination of counts-in-cells statistics with $n_{\mathrm{gal}}$ and
$w_{\mathrm{p}}(r_{\mathrm{p}})$ breaks degeneracies, yielding tighter
constraints on all HOD parameters, and has the potential to definitively
detect or reject assembly bias in existing data. Similarly, Sinha et al.
(2018) and Szewciw et al. (2022) studied a variety of summary statistics and
their constraints on the galaxy–halo connection, and found that group
multiplicity measures, as well as counts-in-cells statistics, are able to
reveal incompleteness in the modelling, and add significantly to the
constraining power. We aim to build off of these analyses.

In this paper, we select galaxy samples based on luminosity and redshift, from
the 7th Data Release of the Sloan Digital Sky Survey (SDSS DR7, Abazajian et
al., 2009), and make independent measurements of our optimal set of
statistics, number density $n_{\mathrm{gal}}$, projected two-point function
$w_{\mathrm{p}}(r_{\mathrm{p}})$, and counts-in-cylinders
$P(N_{\mathrm{CIC}})$. This observable set has several virtues besides the
improvement of constraining power that we demonstrated in W19. All three
statistics have simple configurations, and are easy to measure from data,
without requiring information beyond galaxy positions and luminosities. The
required modelling of these statistics in simulations is likewise minimal.
Both $w_{\mathrm{p}}(r_{\mathrm{p}})$ and $P(N_{\mathrm{CIC}})$ are projected
along the line of sight, and therefore are not particularly sensitive to
redshift measurements or models for the peculiar velocities of galaxies (W19).
As part of our analysis, we conduct careful tests of the impact of
observational effects on our statistics, and verify that our model predictions
are unbiased with respect to data measurements. These tests provide a
systematic evaluation of the reconstruction procedure of these statistics from
a gravity-only $N$-body simulation, which can be generalised to other
statistics and other survey data. We use our measurements to constrain both a
mass-only galaxy–halo connection model, the HOD, and an extension of the HOD
that incorporates adjustable galaxy assembly bias, the dHOD.

The primary goal of this work is to study galaxy assembly bias in the observed
samples using the added information from count statistics. We find that the
inclusion of counts-in-cylinders alongside clustering measurements provides
significant evidence for galaxy assembly bias. Comparing between the
clustering-only analysis and the joint analysis of clustering with counts-in-
cylinders, the latter also reveals the potential incompleteness in the
standard halo mass-only HOD model, and underlines the necessity of properly
modelling galaxy assembly bias. As a subsidiary result, we also produce
tighter constraints on the the full parameter space of the galaxy–halo
connection, which may aid in the interpretation of simulation results and/or
forthcoming survey data.

This paper is organised as follows. The observational data and cosmological
simulation are described in Section 2. In Section 3, we describe our set of
observable statistics, and present measurements from data. We detail our
Markov Chain Monte Carlo fitting in Section 4. We present our constraints on
the galaxy–halo connection, especially the galaxy assembly bias component, in
Section 5. The results are interpreted in Section 6 and discussed in Section
7. Conclusions are drawn in Section 8. In the appendices, we provide technical
details and supplementary results. In Appendix A, we outline the process for
validating the algorithms with which we measure our statistics. We include
supplementary results for alternative data samples in Appendix B.

##  2 Data and Simulation

In this section, we describe the observational data and cosmological
simulation that are used in this work.

###  2.1 Data and Sample Selection

In this work, we use galaxy data from the Sloan Digital Sky Survey Data
Release 7 (SDSS DR7, Abazajian et al., 2009). In particular, we select our
samples from the bright0
catalogue111http://sdss.physics.nyu.edu/lss/dr72/bright/0/, with $r$-band
apparent magnitudes $10.0<m_{r}<17.6$, in the NYU Value-Added Galaxy Catalog
(NYU VAGC, Blanton et al., 2005). The sample contains galaxies that fall
within the survey window, with the bright star-contaminated areas masked out.
We additionally discard sector areas (intersections of tile regions) with low
fractions of galaxies that have spectroscopic redshift measurements, i.e., we
require that the sector fraction $f_{\rm sector}\geq 0.8$. We measure the
resulting angular area of the data footprint to be approximately 7461
$\rm{deg}^{2}$, and the total number of galaxies in the sample is 562620.

Due to the finite size of fibres used in the survey, no two targets on the
same plate can be closer than 55", which results in a fraction of targeted
galaxies not having a measured redshift (known as the fibre collision effect).
Following Zehavi et al. (2011), these galaxies are assigned the redshifts of
their nearest neighbours.

We select volume-limited, luminosity-threshold samples, based on $r$-band
absolute magnitudes that are K-corrected (Blanton & Roweis, 2007) and
passively evolved to the median redshift of the DR7 main galaxy sample,
$z=0.1$. The absolute magnitude values we list as $M_{r}$ throughout this
paper are, in fact, values of $M_{r}-5\log h$ for $h=1$, which are measured
independently of $h$. We apply a universal lower limit of $z_{\rm min}=0.02$
to all of our samples, and adopt the upper bounds of redshift for each
luminosity threshold in Zehavi et al. (2011). In addition to the main samples
which maximise the redshift range for each luminosity threshold, these include
alternative samples for the $M_{r}<-20.0$ and $M_{r}<-19.5$ thresholds with
reduced redshift upper bounds that exclude the Sloan Great Wall. Our sample
selection is illustrated in Figure 1, and the details are listed in Table 1.

Table 1: Volume-limited samples with luminosity threshold. All of the samples have minimum redshift $z_{\rm min}=0.02$. Redshift upper bounds are listed in terms of their products with the speed of light, $c$. The galaxies in each sample satisfy the conditions $z_{\rm min}\leq z_{\rm obs}<z_{\rm max}$ and $M_{r}<M_{r,\rm max}$. The second part of the table lists alternatively selected samples for the $M_{r}<-20.0$ and $M_{r}<-19.5$ thresholds (marked with asterisks), which exclude the cosmic structure known as the Sloan Great Wall by adopting the shallower redshift bounds. $M_{r,\rm max}$ | $cz_{\rm max}[\mathrm{km\ s^{-1}}]$ | $N_{\rm gal}$
---|---|---
| Main samples |
-21.0 | 47650 | 82263
-20.5 | 39700 | 130707
-20.0 | 31900 | 140149
-19.5 | 25450 | 131322
-19.0 | 19250 | 76442
| Alternative samples |
$-20.0^{*}$ | 19250 | 29951
$-19.5^{*}$ | 19250 | 51007
Figure 1: Illustration of our volume-limited, luminosity-threshold galaxy
samples from the NYU VAGC bright0 catalogue. $z_{\rm obs}$ is the observed
redshift, where fibre collided galaxies are assigned the redshifts of their
nearest neighbours. $M_{r}$ is the r-band absolute magnitude. The scatter
points represent the galaxies from the catalogue, and the coloured boxes mark
the selection criteria of the different samples, as are labelled in the
figure. The arrows indicate the alternative $M_{r}<-20.0$ and $M_{r}<-19.5$
samples with lower redshift limits that exclude the Sloan Great Wall.

###  2.2 Simulation

Our forward modelling analyses are based on the Small MultiDark Planck
Simulation (SMDPL), which is a gravitational N-body simulation that belongs to
the series of MultiDark simulations with Planck cosmology (Klypin et al.,
2016). The Small MultiDark Planck Simulation has a cubic volume of side length
400 $h^{-1}\mathrm{Mpc}$, which is comparable to the volume of our
$M_{r}<-21.0$ data sample. The cosmological parameters adopted are
$\Omega_{\Lambda}=0.6929$, $\Omega_{\rm m}=1-\Omega_{\Lambda}=0.3071$,
$\Omega_{\rm b}=0.0482$, $h=0.6777$, $n_{\rm s}=0.96$, and
$\sigma_{8}=0.8228$. We assume this cosmology in our analyses throughout this
paper. The simulation is evolved from $z_{\rm initial}=120$, with $3840^{3}$
particles, implying a particle mass resolution of $m_{\rm p}=9.63\times
10^{7}h^{-1}\mathrm{M}_{\odot}$. We use halo catalogues extracted from the
$z=0.1$ snapshot using the Rockstar halo-finder (Behroozi et al., 2013),
downloaded from https://www.cosmosim.org. We truncate the halo peak mass
$M_{\rm peak}$ at $300\times m_{\rm p}=2.889\times
10^{10}h^{-1}\mathrm{M}_{\odot}$, below which the probability for a halo to
host galaxies above our luminosity thresholds is very low ($\lesssim
10^{-4}$).

##  3 Measurements

In this section, we present measurements of the observable statistics that we
use from the SDSS galaxy samples.

###  3.1 Observable Statistics

We employ summary statistics to extract information from the spatial
distribution of galaxies. In W19, we showed that the combination of the
projected two-point correlation function $w_{\mathrm{p}}(r_{\mathrm{p}})$ and
the counts-in-cylinders statistic $P(N_{\mathrm{CIC}})$ yields tight
constraints on the galaxy–halo connection by breaking degeneracies in the
model parameter space. In this work, we elect to measure these statistics
along with the galaxy number density $n_{\mathrm{gal}}$.

The projected two-point correlation function is defined by

$w_{\mathrm{p}}(r_{\mathrm{p}})=2\int_{0}^{\pi_{\mathrm{max}}}d\pi\
\xi(r_{\mathrm{p}},\pi),$ (1)

where $\xi(r_{\mathrm{p}},\pi)$ is the excess probability of finding galaxy
pairs with projected and line-of-sight separations $r_{\mathrm{p}}$ and $\pi$,
respectively. Considering the depth of the data samples and the size of the
simulation, we choose $\pi_{\mathrm{max}}=40h^{-1}\mathrm{Mpc}$, to include
most correlated pairs and reduce the impact of peculiar velocities, while
excluding distant and less correlated pairs. We compute
$w_{\mathrm{p}}(r_{\mathrm{p}})$ in 12 logarithmically-spaced, radial bins
between $r_{\mathrm{p}}=0.158h^{-1}\mathrm{Mpc}$ and
$r_{\mathrm{p}}=39.81h^{-1}\mathrm{Mpc}$.

The counts-in-cylinders statistic is the probability distribution of the
number of companions found in cylinders around galaxies. As was done in W19,
we centre a cylinder of transverse radius
$r_{\mathrm{CIC}}=2h^{-1}\mathrm{Mpc}$ and line-of-sight half-length
$L=10h^{-1}\mathrm{Mpc}$ on each galaxy in the sample, and count the number of
companion galaxies that fall within the cylinder. We then estimate a
probability distribution of companion number, $P(N_{\mathrm{CIC}})$, which is
the probability that any galaxy has $N_{\mathrm{CIC}}$ companions within the
cylinder. When characterising $P(N_{\mathrm{CIC}})$, we bin $N_{\mathrm{CIC}}$
values linearly on the lower end and logarithmically on the higher end, as
listed in Table 2.

###  3.2 Measurement Algorithm

Our algorithm for measuring the observable set is as follows. The number
density, $n_{\mathrm{gal}}$, is calculated by dividing the number of galaxies
by the volume within the survey footprint and redshift range of each sample.
Both $w_{\mathrm{p}}(r_{\mathrm{p}})$ and $P(N_{\mathrm{CIC}})$ are based on
pair counting, and we first convert the angular and redshift separations
between galaxies to transverse and line-of-sight separations, according to the
cosmological model that we adopt.

For $w_{\mathrm{p}}(r_{\mathrm{p}})$, we use the Landy-Szalay estimator (Landy
& Szalay, 1993),

$\hat{\xi}_{\rm LS}=\frac{\rm{DD}-2\rm{DR}+\rm{RR}}{\rm{RR}},$ (2)

where DD is the normalised galaxy–galaxy pair count, RR is the normalised
random–random pair count, and DR is the normalised cross pair count between
galaxies and randoms. Randoms are drawn from a uniform distribution in the
redshift range and footprint of the survey, to account for the geometry of the
survey volume. $\xi$ is integrated along the line of sight to
$\pi_{\mathrm{max}}$. The calculation is done using the Corrfunc package
(Sinha, 2016).

For $P(N_{\mathrm{CIC}})$, we impose additional criteria on the galaxies used
as cylinder centres, such that the sampling of companions is sufficient in the
neighbourhood of each cylinder centre. We define the angular completeness
$f_{\rm AC}$ of a galaxy to be the fraction of the circular area with radius
$r_{\mathrm{CIC}}=2h^{-1}\mathrm{Mpc}$ around it that falls inside the survey
footprint. For cylinder centres we require that the angular completeness
around them to be above 0.9, and cylinders centred on them to be completely
within the redshift ranges of the volume-limited samples. For each of the
cylinders that satisfy these requirements, we then count companions that
belong to the same volume-limited, luminosity-threshold sample. We upscale the
count numbers by $1/f_{\rm AC}$ to account for any angular incompleteness,
which results in non-integer counts. The counts are binned to yield the
probability distribution, $P(N_{\mathrm{CIC}})$.

For all the measurements, we adopt the nearest-neighbour corrected redshifts
for fibre collided galaxies that do not have spectroscopic redshifts.

###  3.3 Covariance Estimation

We calculate the jackknife covariance, which provides an estimate of the
uncertainty due to the finite volume of the survey. We use the iterative
method described in Zhou et al. (2021)222Code available at
https://github.com/rongpu/pixel_partition, to divide the survey footprint into
subareas of similar sizes. We show a map of the cells in Figure 2, where
galaxies in different cell are plotted in different colours.

Figure 2: SDSS footprint divided into jackknife cells. Objects in different
cell are plotted in different colours. All the cells are similar in size,
except some in the three stripes away from the main footprint, which are
smaller than average.

For each sample, we exclude one cell at a time in our measurement to obtain an
ensemble of jackknife subsamples. In particular, for $n_{\mathrm{gal}}$,
galaxies are excluded if they fall in the excluded cell, for
$w_{\mathrm{p}}(r_{\mathrm{p}})$, we exclude a pair of objects if either or
both are in the excluded cell, whereas for $P(N_{\mathrm{CIC}})$, we keep all
the galaxies as companions, and only exclude cylinder centres that fall in the
excluded cell from the probability distribution. We then record the covariance
matrix across the ensemble as our jackknife covariance.

###  3.4 Measurement Results

The $w_{\mathrm{p}}(r_{\mathrm{p}})$ values we measure from data are shown in
Figure 3 in the form of $r_{\mathrm{p}}\times w_{\mathrm{p}}$, where each
luminosity threshold is plotted individually in the first five panels, and
compared alongside each other in the last panel. The error bars show the
jackknife error of each $r_{\mathrm{p}}$ bin. For the $M_{r}<-20.0$ and
$M_{r}<-19.5$ thresholds, the alternative samples with the shallower redshift
ranges are plotted in grey. Our measurements for
$w_{\mathrm{p}}(r_{\mathrm{p}})$ are consistent with the measurements in
Zehavi et al. (2011). Number densities are also shown in text in the
corresponding panels, with jackknife errors included in parentheses.
$n_{\mathrm{gal}}$ values for the alternative samples are marked with
asterisks. The $P(N_{\mathrm{CIC}})$ statistics are similarly shown in Figure
4. The values of the statistics are listed in Table 2.

Brighter samples have lower number densities, but generally stronger
clustering. The $P(N_{\mathrm{CIC}})$ values in all the bins sum up to unity,
as $P(N_{\mathrm{CIC}})$ is a probability distribution. Brighter galaxies are
rarer, and tend to have fewer companions, resulting in higher probabilities of
smaller $N_{\mathrm{CIC}}$, and lower probabilities of larger
$N_{\mathrm{CIC}}$. The samples that have smaller volumes have larger
jackknife errors for all of the observables.

For the $M_{r}<-20.0$ and $M_{r}<-19.5$ luminosity thresholds, the alternative
redshift limit that excludes the Sloan Great Wall results in lower number
densities and weaker large-scale clustering, as is expected. For both
thresholds, excluding the Great Wall increases the probability of galaxies
having only a few companions in cylinders, reduces that of intermediate
numbers of companions, but shows an increase in the probability of having very
large numbers of companions. However, neither the increase of
$P(N_{\mathrm{CIC}})$ at the large $N_{\mathrm{CIC}}$ end nor the increase of
$w_{\mathrm{p}}(r_{\mathrm{p}})$ are statistically significant.

Figure 3: Measurement of the galaxy number density $n_{\mathrm{gal}}$ and the projected two-point function $w_{\mathrm{p}}$ from SDSS data. The two point function is shown in the form of $r_{\mathrm{p}}\times w_{\mathrm{p}}$ instead of $w_{\mathrm{p}}$ for legibility. Each of the first five panels shows the measurement for one luminosity sample with jackknife error bars that represent the cosmic variance. For the $-20.0$ and $-19.5$ thresholds, results for the alternative samples without the Sloan Great Wall are shown in the respective panels as values with asterisks and grey lines. The bottom right panel shows $r_{\mathrm{p}}\times w_{\mathrm{p}}(r_{\mathrm{p}})$ of the five thresholds together, with vertical offsets of 0.25 dex each, starting from the $-20.0$ sample, for visual clarity. Figure 4: Measurement of the counts-in-cylinders statistic $P(N_{\mathrm{CIC}})$ from SDSS data. Similar to Figure 3, each of the first five panels shows the measurement for one luminosity sample with jackknife error bars that represent the cosmic variance. The statistic is represented as the probability distribution of $N_{\mathrm{CIC}}$, normalised by the bin widths. For the $-20.0$ and $-19.5$ thresholds, results for the alternative samples without the Sloan Great Wall are shown in the respective panels as grey lines. The bottom right panel shows the $P(N_{\mathrm{CIC}})$ of the five thresholds together, with no vertical offset. Table 2: Measured values of the observable statistics from SDSS data. $n_{\mathrm{gal}}$, $w_{\mathrm{p}}(r_{\mathrm{p}})$, and $P(N_{\mathrm{CIC}})$ measurements are listed in three respective sections from top to bottom. The number density values $n_{\mathrm{gal}}$ are listed in units of $10^{-3}h^{3}{\rm Mpc}^{-3}$. The leftmost column shows the $r_{\mathrm{p}}$ bin centre in units of $h^{-1}\mathrm{Mpc}$ for $w_{\mathrm{p}}$, and the edges of each bin in $N_{\mathrm{CIC}}$ for $P(N_{\mathrm{CIC}})$. The values listed for $w_{\mathrm{p}}(r_{\mathrm{p}})$ are in units of $h^{-1}\mathrm{Mpc}$, and the values listed for $P(N_{\mathrm{CIC}})$ are the probability in each bin, which would sum up to unity. Jackknife errors are shown in brackets. | $M_{r}<-21.0$ | $M_{r}<-20.5$ | $M_{r}<-20.0$ | $M_{r}<-19.5$ | $M_{r}<-19.0$ | $M_{r}<-20.0^{*}$ | $M_{r}<-19.5^{*}$
---|---|---|---|---|---|---|---
|  $n_{\mathrm{gal}}\left[10^{-3}h^{3}{\rm Mpc}^{-3}\right]$
| 1.128(0.021) | 3.044(0.063) | 6.192(0.155) | 11.32(0.279) | 15.28(0.506) | 5.985(0.214) | 10.19(0.349)
$r_{\mathrm{p}}\left[h^{-1}\mathrm{Mpc}\right]$ |  $w_{\mathrm{p}}(r_{\mathrm{p}})\left[h^{-1}\mathrm{Mpc}\right]$
0.200 | 508.36(13.00) | 365.33(7.38) | 311.92(8.25) | 268.89(8.67) | 276.92(16.64) | 308.88(17.98) | 288.03(17.14)
0.316 | 332.75(8.30) | 251.26(4.73) | 224.05(6.71) | 195.09(7.68) | 198.23(14.26) | 216.09(14.45) | 207.49(14.93)
0.501 | 201.96(4.74) | 163.95(4.17) | 154.16(5.82) | 134.95(6.13) | 139.73(12.11) | 148.19(12.91) | 144.09(12.32)
0.794 | 134.25(3.51) | 111.89(3.23) | 106.07(4.84) | 92.18(4.97) | 95.93(9.47) | 103.46(9.92) | 99.67(9.62)
1.259 | 85.17(2.32) | 75.32(2.58) | 70.86(3.78) | 60.96(3.72) | 63.29(6.79) | 68.98(6.96) | 65.68(6.67)
1.995 | 59.77(1.76) | 53.23(2.04) | 50.78(3.06) | 42.72(3.13) | 43.15(5.57) | 47.56(5.88) | 45.20(5.55)
3.162 | 43.54(1.51) | 38.23(1.62) | 36.87(2.46) | 30.64(2.47) | 30.89(4.32) | 34.55(4.74) | 32.29(4.31)
5.012 | 30.86(1.13) | 26.89(1.35) | 26.36(2.04) | 21.87(2.07) | 21.65(3.29) | 23.78(3.38) | 22.65(3.34)
7.943 | 20.96(1.00) | 18.39(1.21) | 18.47(1.66) | 14.80(1.79) | 13.86(2.50) | 15.16(2.61) | 14.27(2.54)
12.59 | 12.35(0.81) | 11.21(1.03) | 11.84(1.32) | 9.27(1.42) | 8.03(1.65) | 8.77(1.91) | 8.48(1.74)
19.95 | 6.75(0.70) | 6.26(0.91) | 6.92(1.11) | 5.07(1.26) | 3.72(1.38) | 3.67(1.51) | 3.75(1.41)
31.62 | 3.37(0.63) | 3.50(0.86) | 4.35(1.09) | 2.97(1.15) | 2.63(1.45) | 2.53(1.58) | 2.73(1.52)
$N_{\mathrm{CIC}}$ |  $P(N_{\mathrm{CIC}})$
-0.5, 0.5 | 0.379(4.82e-3) | 0.189(3.87e-3) | 0.096(2.82e-3) | 0.049(1.82e-3) | 0.032(1.79e-3) | 0.106(4.95e-3) | 0.054(2.97e-3)
0.5, 1.5 | 0.249(2.49e-3) | 0.187(2.80e-3) | 0.117(2.91e-3) | 0.071(2.48e-3) | 0.049(2.53e-3) | 0.123(5.43e-3) | 0.077(3.82e-3)
1.5, 2.5 | 0.149(2.07e-3) | 0.148(1.87e-3) | 0.113(2.43e-3) | 0.078(2.47e-3) | 0.061(2.98e-3) | 0.117(4.86e-3) | 0.086(3.51e-3)
2.5, 3.5 | 0.085(1.76e-3) | 0.111(1.49e-3) | 0.099(1.96e-3) | 0.077(2.22e-3) | 0.060(2.87e-3) | 0.106(3.92e-3) | 0.081(3.22e-3)
3.5, 4.5 | 0.052(1.62e-3) | 0.083(1.27e-3) | 0.084(1.59e-3) | 0.072(2.04e-3) | 0.061(2.50e-3) | 0.084(2.93e-3) | 0.077(3.25e-3)
4.5, 5.5 | 0.029(1.21e-3) | 0.063(1.19e-3) | 0.069(1.14e-3) | 0.068(1.75e-3) | 0.055(2.24e-3) | 0.068(2.50e-3) | 0.068(2.47e-3)
5.5, 6.5 | 0.021(1.08e-3) | 0.046(1.05e-3) | 0.058(1.05e-3) | 0.058(1.31e-3) | 0.052(2.30e-3) | 0.055(2.87e-3) | 0.056(2.01e-3)
6.5, 7.5 | 0.014(8.48e-4) | 0.036(1.09e-3) | 0.049(1.06e-3) | 0.050(1.03e-3) | 0.047(1.76e-3) | 0.048(2.42e-3) | 0.050(1.88e-3)
7.5, 8.5 | 0.008(5.99e-4) | 0.027(8.79e-4) | 0.040(9.48e-4) | 0.043(1.05e-3) | 0.045(1.60e-3) | 0.039(2.25e-3) | 0.041(1.61e-3)
8.5, 9.5 | 0.005(4.64e-4) | 0.021(8.50e-4) | 0.036(9.49e-4) | 0.039(8.89e-4) | 0.038(1.54e-3) | 0.036(2.36e-3) | 0.037(1.59e-3)
9.5, 11.5 | 0.005(5.55e-4) | 0.030(1.38e-3) | 0.055(1.60e-3) | 0.067(1.70e-3) | 0.066(1.98e-3) | 0.050(2.98e-3) | 0.065(2.93e-3)
11.5, 12.5 | 0.002(2.56e-4) | 0.010(6.21e-4) | 0.021(7.80e-4) | 0.028(8.75e-4) | 0.030(1.29e-3) | 0.023(2.38e-3) | 0.027(1.41e-3)
12.5, 14.5 | 0.001(3.72e-4) | 0.016(1.10e-3) | 0.035(1.36e-3) | 0.048(1.53e-3) | 0.053(2.28e-3) | 0.033(2.71e-3) | 0.047(2.61e-3)
14.5, 17.5 | 0.001(2.71e-4) | 0.013(1.05e-3) | 0.037(1.69e-3) | 0.055(2.01e-3) | 0.063(3.12e-3) | 0.028(2.97e-3) | 0.052(3.21e-3)
17.5, 19.5 | 0.000(1.35e-4) | 0.006(5.12e-4) | 0.018(1.07e-3) | 0.028(1.28e-3) | 0.034(1.93e-3) | 0.014(1.75e-3) | 0.027(2.26e-3)
19.5, 23.5 | 0.000(6.37e-5) | 0.008(8.23e-4) | 0.024(1.83e-3) | 0.040(1.74e-3) | 0.052(3.00e-3) | 0.022(2.99e-3) | 0.036(2.44e-3)
23.5, 26.5 | 0 | 0.003(4.13e-4) | 0.012(1.13e-3) | 0.022(1.25e-3) | 0.030(2.39e-3) | 0.012(2.00e-3) | 0.022(1.90e-3)
26.5, 30.5 | 0 | 0.002(3.80e-4) | 0.011(1.13e-3) | 0.025(1.69e-3) | 0.033(2.35e-3) | 0.009(1.72e-3) | 0.023(2.80e-3)
30.5, 35.5 | 0 | 0.001(2.83e-4) | 0.009(1.09e-3) | 0.021(1.63e-3) | 0.028(2.19e-3) | 0.010(2.49e-3) | 0.017(2.24e-3)
35.5, 41.5 | 0 | 0.000(1.19e-4) | 0.007(9.51e-4) | 0.018(1.66e-3) | 0.027(2.45e-3) | 0.010(2.49e-3) | 0.015(2.55e-3)
41.5, 47.5 | 0 | 0.000(1.21e-4) | 0.004(7.29e-4) | 0.012(1.38e-3) | 0.019(2.42e-3) | 0.004(1.66e-3) | 0.010(1.92e-3)
47.5, 54.5 | 0 | 0.000(1.87e-5) | 0.003(6.83e-4) | 0.010(1.30e-3) | 0.013(2.24e-3) | 0.003(1.55e-3) | 0.009(2.09e-3)
54.5, 63.5 | 0 | 0 | 0.001(4.34e-4) | 0.008(1.21e-3) | 0.014(2.20e-3) | 0.002(1.26e-3) | 0.008(2.33e-3)
63.5, 73.5 | 0 | 0 | 0.000(2.22e-4) | 0.005(9.92e-4) | 0.012(2.46e-3) | 0.001(4.72e-4) | 0.005(1.77e-3)
73.5, 84.5 | 0 | 0 | 0.000(2.32e-4) | 0.003(8.14e-4) | 0.009(2.32e-3) | 0 | 0.003(1.36e-3)
84.5, 97.5 | 0 | 0 | 0.000(1.43e-4) | 0.002(7.83e-4) | 0.006(1.91e-3) | 0 | 0.003(1.57e-3)
97.5, 112.5 | 0 | 0 | 0 | 0.001(4.39e-4) | 0.003(1.24e-3) | 0 | 0.001(8.22e-4)
112.5, 129.5 | 0 | 0 | 0 | 0.000(7.32e-5) | 0.003(1.41e-3) | 0 | 0.000(1.98e-4)
129.5, 149.5 | 0 | 0 | 0 | 0 | 0.003(1.59e-3) | 0 | 0

##  4 Fitting Models to Observational Data

In this section, we describe how we construct our models and the approach for
fitting the clustering and counts-cylinders measurements using galaxy–halo
connection models with and without galaxy assembly bias components.

###  4.1 Galaxy–Halo Connection Models

We fit two galaxy–halo connection models –– one halo mass-only model, in which
the galaxy content of a halo depends only upon the mass of the halo, and an
alternative model that incorporates galaxy assembly bias –– to the data
measurement. We choose the halo occupation distribution (HOD; Zheng et al.,
2007) model for the former, and the decorated halo occupation distribution
(dHOD; Hearin et al., 2016) model for the latter. We give a brief description
of the models in this subsection, and encourage the interested reader to refer
back to the original papers, as well as Section 2.2 of W19, for details and
motivations for the specific choices we make.

In the standard HOD, the number of galaxies in a given halo is determined
solely by the mass of the halo, $M_{\mathrm{vir}}$. The dependence of the mean
occupation on halo mass is modulated by 5 parameters. The mean number of
central and satellite galaxies in halos of fixed mass are modelled separately
as

$\displaystyle\langle N_{\mathrm{cen}}|M_{\mathrm{vir}}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(1+\mathrm{erf}\left[\frac{\log
M_{\mathrm{vir}}-\log
M_{\mathrm{min}}}{\sigma_{\mathrm{log}M}}\right]\right),$ (3)
$\displaystyle\langle N_{\mathrm{sat}}|M_{\mathrm{vir}}\rangle$
$\displaystyle=$
$\displaystyle\left(\frac{M_{\mathrm{vir}}-M_{0}}{M_{1}}\right)^{\alpha}\times\langle
N_{\mathrm{cen}}|M_{\mathrm{vir}}\rangle.$ (4)

The number of central galaxies assigned to any particular halo,
$N_{\mathrm{cen}}$, is drawn from a Bernoulli distribution with a mean given
by Eq. (3). Therefore, any individual halo can have, at most, one central
galaxy. The number of satellite galaxies assigned to any halo,
$N_{\mathrm{sat}}$, is drawn from a Poisson distribution with mean given by
Eq. (4). While the probability for a halo to host satellite galaxies is
modulated by the mean central occupation, a central galaxy is not strictly
required for satellites to be present for individual haloes.

The decorated HOD has an additional galaxy assembly bias component. This is
implemented by introducing a dependence of the mean numbers of satellite and
central galaxies on a secondary halo property $x$ (any property aside from
mass is a secondary property). At each halo mass, we categorise the haloes
into two sub-populations, with a pivot value of $x_{\mathrm{piv}}$,

$\displaystyle\langle
N_{\mathrm{gal}}|M_{\mathrm{vir}},x>x_{\mathrm{piv}}\rangle$ $\displaystyle=$
$\displaystyle\langle N_{\mathrm{gal}}|M_{\mathrm{vir}}\rangle+\delta
N_{\mathrm{gal}},$ (5) $\displaystyle\langle
N_{\mathrm{gal}}|M_{\mathrm{vir}},x\leq x_{\mathrm{piv}}\rangle$
$\displaystyle=$ $\displaystyle\langle
N_{\mathrm{gal}}|M_{\mathrm{vir}}\rangle-\delta N_{\mathrm{gal}},$ (6)

where $N_{\mathrm{gal}}$ is either $N_{\mathrm{cen}}$ or $N_{\mathrm{sat}}$.
In this work, we choose the NFW (Navarro et al., 1997) halo concentration
$c_{\mathrm{vir}}$ as the secondary halo property, that is,
$x=c_{\mathrm{vir}}$, and choose $x_{\mathrm{piv}}$ to be the median value of
$c_{\mathrm{vir}}$ in each mass bin. The variation is then given by

$\displaystyle\delta N_{\mathrm{cen}}$ $\displaystyle=$ $\displaystyle
A_{\mathrm{cen}}\,\mathrm{min}\left[\langle
N_{\mathrm{cen}}|M_{\mathrm{vir}}\rangle,1-\langle
N_{\mathrm{cen}}|M_{\mathrm{vir}}\rangle\right],$ (7) $\displaystyle\delta
N_{\mathrm{sat}}$ $\displaystyle=$ $\displaystyle A_{\mathrm{sat}}\,\langle
N_{\mathrm{sat}}|M_{\mathrm{vir}}\rangle,$ (8)

where the two assembly bias parameters, $A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$, range from -1 to 1, and the special case of
$A_{\mathrm{cen}}=0$ and $A_{\mathrm{sat}}=0$ is equivalent to the standard
HOD (i.e., no assembly bias). Throughout the text, we refer to
$A_{\mathrm{cen}}$ as the central assembly bias parameter, and
$A_{\mathrm{sat}}$ as the satellite assembly bias parameter. The existence of
central and satellite assembly biases (or the lack thereof) discussed in this
work is limited to this specific model choice.

For the purposes of this study, we assign galaxies positions and velocities
relative to their halo centres in the default manner specified within the
halotools software. In particular, central galaxies are placed at the centres
of their host haloes and inherit the host halo’s peculiar velocity as
identified by the halo finder. Satellite galaxies are distributed according to
a spherically-symmetric NFW profile with the same concentration as the dark
matter distribution. Satellite peculiar velocities are modelled assuming
isotropy, and the radial velocity follows a Gaussian distribution, whose first
moment is the host halo velocity, and second moment is the solution of the
isotropic Jeans equation (Klypin et al., 1999), for an NFW profile with the
same concentration as the dark matter in the halo.

###  4.2 Model Prediction of Statistics

We predict $n_{\mathrm{gal}}$, $w_{\mathrm{p}}(r_{\mathrm{p}})$, and
$P(N_{\mathrm{CIC}})$ from the galaxy–halo connection models using the $z=0.1$
snapshot of the cubic SMDPL simulation. The details of the measurement of
these statistics are described in Section A.4. To ensure that the predictions
from the cubic simulation volume are consistent with observational estimations
of the statistics, we need to understand the observational effects that enter
the SDSS data measurements. Apart from the corrections on absolute magnitudes
already made in the NYU VAGC, we consider major observational effects in SDSS
(see Section 2.1 for details) including i) the light cone geometry with a non-
trivial footprint, ii) the fibre collision effect and other failures to
retrieve galaxy redshifts, and iii) redshift space distortion due to peculiar
velocities of galaxies. While iii) can be incorporated into the cubic
simulation by modelling mock galaxy velocities, i) and ii) are unique to the
geometry of observational data. We construct cone mocks that incorporate these
observational effects, and use them to validate our measurement algorithms,
demonstrating that our statistic estimates from the observational data and the
cubic simulation are consistent within error. The details of the cone
construction and the validation process are described in Appendix A.

###  4.3 Covariance Matrix

In fitting data measurements with mock measurements from the simulation, we
need to account for both the uncertainty in the observation and the
uncertainty in the mock estimation. Therefore, the covariance matrix that we
use for the fitting is the sum of the data component and the theory component.
In Figure 5, we show the normalised covariance matrix for the $M_{r}<-19.0$
sample as an example. The two panels on the left are the jackknife covariances
from the SDSS data and the SMDPL mock separately, and the rightmost panel is
the sum of both. The estimation of the SDSS data covariance is detailed in
Section 3.3, and the estimation of the SMDPL mock covariance is detailed in
Section A.4.2. In plotting the matrices, they are normalised by the diagonal
elements, such that all the diagonal elements are normalised to 1 by
construction, and the off-diagonal elements range between -1 and 1, which are
colour coded in the figure. The data covariance is noisier due to its smaller
volume than the simulation covariance.

We treat the set of observables as one data vector, and include covariance
terms across observables. For each luminosity sample, in both the data and the
mock components of the covariance, $n_{\mathrm{gal}}$ is positively correlated
with $w_{\mathrm{p}}$ in general and positively correlated with counts of
companions in cylinders. Within the same galaxy population, galaxy clustering
is stronger in regions of space that are slightly denser than average. The
correlations between $w_{\mathrm{p}}(r_{\mathrm{p}})$ values across different
scales are positive within the range we investigate, and their correlations
with $P(N_{\mathrm{CIC}})$ approximately follow those of $n_{\mathrm{gal}}$.
On the other hand, $P(N_{\mathrm{CIC}})$ values at the higher and lower ends
are anti-correlated by construction, as these probabilities sum up to unity.

Figure 5: Normalised covariance matrices (i.e., correlation matrices) for the
$M_{r}<-19.0$ sample. The left panel is the jackknife covariance of the SDSS
data sample, the middle panel that of the SMDPL simulation, and the right
panel shows the sum of the two. In the figure, the $r_{\mathrm{p}}$ bins and
the $N_{\mathrm{CIC}}$ bins both increase from left to right and from bottom
to top. The diagonal elements of the correlation matrices are 1 by
construction.

###  4.4 MCMC Fitting

We use the Markov Chain Monte Carlo (MCMC) method to infer parameter
constraints from the measurements. The covariance matrices we use in the fit
include the cross covariances between different statistics, as is described in
Section 4.3. When sampling the parameter space, we populate the haloes in the
simulation with galaxies according to each parameter set, make mock
measurements on the resulting galaxy catalogue, and compute the corresponding
likelihood. We assume likelihood $\mathcal{L}\propto e^{-\chi^{2}/2}$, with

$\chi^{2}=\sum_{i,j}\Delta f_{i}[\mathbf{C}^{-1}]_{ij}\Delta f_{j},$ (9)

where $f_{i}$ and $f_{j}$ are the $i$th and $j$th element of the joint
statistic vector $\mathbf{f}$, and $\mathbf{C}$ is the full covariance.
Because the three types of statistics, $n_{\mathrm{gal}}$,
$w_{\mathrm{p}}(r_{\mathrm{p}})$, and $P(N_{\mathrm{CIC}})$, have a wide range
of values, we calculate the Moore-Penrose pseudo-inversion (Penrose, 1955) of
the covariance. In performing the pseudo-inversion, we choose a conservative
cutoff of $10^{-15}$ on eigenvalues, but the results are not sensitive to the
specific choice. We do not account for the Hartlap factor (Hartlap et al.,
2007) in the inversion, as the effective number of data points is not well
defined due to the correlation between the data bins. This choice does not
affect our parameter constraints, but may have a moderate effect on the
absolute values of $\chi^{2}$. In addition, the covariance matrix itself
depends upon the parameters of the model; however, we neglect this dependence
as covariances tend to be relatively weakly varying functions of the model
parameters (Eifler et al., 2009). We assume uniform priors within certain
intervals on the parameters, listed in Table 3.

Table 3: Prior intervals adopted for MCMC fits. The $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ priors only apply to the decorated HOD fits, whereas in the standard HOD model both parameters are fixed to 0. Parameter | Prior interval
---|---
$\log M_{\mathrm{min}}$ | [11.0, 14.0]
$\sigma_{\mathrm{log}M}$ | [0.02, 1.5]
$\alpha$ | [0, 1.5]
$\log M_{1}$ | [11.5, 15.0]
$\log M_{0}$ | [9.0, 14.0]
$A_{\mathrm{cen}}$ | [-1, 1]
$A_{\mathrm{sat}}$ | [-1, 1]

With the above likelihood and prior, we use the emcee (Foreman-Mackey et al.,
2013) package, which implements an affine-invariant ensemble sampler (Goodman
& Weare, 2010), to sample from the posterior distribution in the parameter
space. We use the TabCorr333Available at
https://github.com/johannesulf/TabCorr. package, which is based on the
tabulating method proposed in Zheng & Guo (2016), to expedite the computation
of $w_{\mathrm{p}}(r_{\mathrm{p}})$.

##  5 Results

In this section, we report the constraints we get on the galaxy–halo
connection, making comparisons between the different statistic sets, models,
and samples.

The galaxy samples we analyse are selected from SDSS DR7. These include the
five main samples listed in Table 1, that maximise sample depths for the
$M_{r}<-21.0$, $M_{r}<-20.5$, $M_{r}<-20.0$, $M_{r}<-19.5$, and $M_{r}<-19.0$
luminosity thresholds. The results for the alternative $M_{r}<-20.0$ and
$M_{r}<-19.5$ samples that exclude the Sloan Great Wall will be shown in
Appendix B. For each SDSS galaxy sample, we consider four fitting cases:

  1. 1.

the standard HOD inferred from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$;

  2. 2.

the standard HOD inferred from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$;

  3. 3.

the decorated HOD inferred from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$; and

  4. 4.

the decorated HOD inferred from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$.

###  5.1 Marginalised Constraints on Assembly Bias Parameters

We first present the 1-D marginalised constraints on each HOD and dHOD
parameter for the five main samples and the four fitting cases. In Table 4, we
list the inferred parameter values, quoting the medians of the marginalised
distributions as central values, with asymmetric “1$\sigma$” errors that
correspond to the 16th and 84th percentile bounds.

Our primary scientific focus is on the assembly bias parameters and we
therefore begin our discussion with those parameters. As is evident from the
values in the seventh column of Table 4, there are several samples for which
clustering data suggest positive central galaxy assembly bias and for which
the addition of CIC data provide further support for positive central galaxy
assembly bias. For the $M_{r}<-20.5$, $M_{r}<-19.5$, and $M_{r}<-19.0$
samples, the probability that $A_{\mathrm{cen}}\leq 0$ is $3.0\%$, $0.1\%$,
and $6.4\%$ respectively. For the $M_{r}<-20.0$ sample, cases with
$A_{\mathrm{cen}}\leq 0$ are not detected with the current length of the
chain, and the probability that $A_{\mathrm{cen}}\leq 0.3$ is $0.1\%$.
Positive values of $A_{\mathrm{cen}}$ correspond to the case in which galaxies
are more likely to reside within high-concentration haloes and our analysis
reveals intriguing indications that this may be the case for several of the
SDSS DR7 luminosity threshold samples. In contrast to central galaxy assembly
bias, there is no strong evidence of satellite galaxy assembly bias.

We highlight the constraints on the assembly bias parameters in Figure 6.
1$\sigma$ and 2$\sigma$ error bars are shown for the central and satellite
assembly bias parameters, $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$,
separately, in the two panels. For each parameter and each galaxy sample, the
figure compares the constraints from the clustering-only fits
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$”) and the fits that also
include CIC
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$+$P(N_{\mathrm{CIC}})$”).
Notice that the addition of companion counts in cylinders not only
significantly reduces the confidence ranges on the inferred assembly bias
parameters, $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$, but also bolsters the
evidence for $A_{\mathrm{cen}}>0$ for all threshold samples. This in turn
strengthens our ability to distinguish between the standard mass-only HOD and
the decorated HOD that includes galaxy assembly bias. We will discuss this
result in more detail, in Section 6.1 and Section 6.2.

Table 4: Inferred constraints on HOD and dHOD parameters. For each sample, we tabulate the constraints on the parameters for the four cases listed in Section 5. We quote the median of the marginalised distribution of each parameter as our fitting result, and the bounds of the asymmetric error bars correspond to the 16% and 84% percentiles. There are five parameters for the HOD model, and seven for the decorated HOD. $\log M_{\mathrm{min}}$ and $\sigma_{\mathrm{log}M}$ are parameters that control the central galaxy occupation, and $\alpha$, $\log M_{1}$ and $\log M_{0}$ control the satellite occupation. $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ are the galaxy assembly bias parameters for centrals and satellites respectively, and are only applicable to the decorated HOD model. $M_{\mathrm{min}}$ is the mass at which a halo has a 50% probability to host a central galaxy; $\sigma_{\mathrm{log}M}$ determines the steepness of the central occupation’s transition from zero to unity; $\alpha$ is the index of the satellite occupation power law; $M_{1}$ indicates the halo mass at which the mean satellite number is one; and $M_{0}$ is the mass below which there can be no centrals. $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ can vary independently of each other, both ranging from -1 to 1, and in the absence of assembly bias, they assume values of 0.  $M_{r}<-21.0$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
---|---|---|---|---|---|---|---
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$12.730^{+0.113}_{-0.064}$ | $0.358^{+0.212}_{-0.232}$ | $1.142^{+0.102}_{-0.210}$ | $13.945^{+0.054}_{-0.134}$ | $12.622^{+0.508}_{-2.144}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$12.679^{+0.059}_{-0.027}$ | $0.228^{+0.164}_{-0.157}$ | $0.938^{+0.086}_{-0.092}$ | $13.817^{+0.066}_{-0.075}$ | $13.113^{+0.131}_{-0.180}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$12.716^{+0.123}_{-0.054}$ | $0.318^{+0.242}_{-0.209}$ | $1.103^{+0.115}_{-0.188}$ | $13.979^{+0.049}_{-0.118}$ | $12.469^{+0.589}_{-2.130}$ | $-0.009^{+0.705}_{-0.686}$ | $-0.273^{+0.673}_{-0.515}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$12.671^{+0.063}_{-0.020}$ | $0.185^{+0.198}_{-0.123}$ | $0.892^{+0.093}_{-0.094}$ | $13.844^{+0.091}_{-0.088}$ | $13.086^{+0.152}_{-0.202}$ | $0.105^{+0.619}_{-0.702}$ | $-0.235^{+0.444}_{-0.512}$
$M_{r}<-20.5$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$12.258^{+0.049}_{-0.023}$ | $0.172^{+0.172}_{-0.108}$ | $1.095^{+0.061}_{-0.069}$ | $13.544^{+0.050}_{-0.059}$ | $12.264^{+0.289}_{-1.223}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$12.246^{+0.016}_{-0.011}$ | $0.105^{+0.085}_{-0.059}$ | $0.947^{+0.033}_{-0.034}$ | $13.432^{+0.033}_{-0.036}$ | $12.681^{+0.078}_{-0.089}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$12.325^{+0.166}_{-0.075}$ | $0.402^{+0.278}_{-0.254}$ | $1.041^{+0.070}_{-0.073}$ | $13.528^{+0.054}_{-0.065}$ | $12.267^{+0.305}_{-1.053}$ | $0.783^{+0.165}_{-0.453}$ | $-0.219^{+0.277}_{-0.284}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$12.265^{+0.031}_{-0.021}$ | $0.218^{+0.099}_{-0.104}$ | $0.950^{+0.042}_{-0.045}$ | $13.454^{+0.039}_{-0.038}$ | $12.617^{+0.096}_{-0.125}$ | $0.811^{+0.141}_{-0.295}$ | $-0.147^{+0.158}_{-0.226}$
$M_{r}<-20.0$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$11.953^{+0.091}_{-0.032}$ | $0.238^{+0.243}_{-0.150}$ | $1.018^{+0.063}_{-0.065}$ | $13.180^{+0.073}_{-0.081}$ | $12.393^{+0.234}_{-0.398}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$11.937^{+0.018}_{-0.014}$ | $0.109^{+0.094}_{-0.066}$ | $0.902^{+0.031}_{-0.032}$ | $13.072^{+0.040}_{-0.046}$ | $12.552^{+0.085}_{-0.088}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$12.167^{+0.401}_{-0.204}$ | $0.690^{+0.450}_{-0.396}$ | $0.955^{+0.069}_{-0.080}$ | $13.144^{+0.081}_{-0.110}$ | $12.382^{+0.288}_{-0.676}$ | $0.826^{+0.130}_{-0.301}$ | $-0.211^{+0.339}_{-0.319}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$11.973^{+0.027}_{-0.020}$ | $0.298^{+0.058}_{-0.062}$ | $0.928^{+0.033}_{-0.039}$ | $13.118^{+0.043}_{-0.050}$ | $12.420^{+0.113}_{-0.123}$ | $0.922^{+0.060}_{-0.138}$ | $-0.166^{+0.140}_{-0.184}$
$M_{r}<-19.5$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$11.806^{+0.453}_{-0.144}$ | $0.547^{+0.592}_{-0.389}$ | $1.049^{+0.041}_{-0.058}$ | $12.975^{+0.047}_{-0.077}$ | $11.599^{+0.677}_{-1.703}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$11.751^{+0.034}_{-0.032}$ | $0.443^{+0.069}_{-0.076}$ | $0.913^{+0.033}_{-0.036}$ | $12.817^{+0.048}_{-0.057}$ | $12.449^{+0.107}_{-0.120}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$11.940^{+0.432}_{-0.262}$ | $0.762^{+0.510}_{-0.530}$ | $0.993^{+0.065}_{-0.071}$ | $12.947^{+0.078}_{-0.098}$ | $11.655^{+0.676}_{-1.669}$ | $0.605^{+0.282}_{-0.516}$ | $-0.240^{+0.552}_{-0.409}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$11.709^{+0.029}_{-0.023}$ | $0.367^{+0.066}_{-0.064}$ | $0.890^{+0.040}_{-0.049}$ | $12.785^{+0.054}_{-0.059}$ | $12.527^{+0.100}_{-0.111}$ | $0.711^{+0.194}_{-0.243}$ | $-0.087^{+0.185}_{-0.369}$
$M_{r}<-19.0$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$11.733^{+0.395}_{-0.158}$ | $0.594^{+0.550}_{-0.404}$ | $1.046^{+0.031}_{-0.036}$ | $12.838^{+0.036}_{-0.044}$ | $10.477^{+1.064}_{-1.005}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$11.597^{+0.041}_{-0.048}$ | $0.410^{+0.087}_{-0.149}$ | $0.894^{+0.050}_{-0.042}$ | $12.649^{+0.071}_{-0.072}$ | $12.328^{+0.130}_{-0.185}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$11.797^{+0.390}_{-0.218}$ | $0.710^{+0.525}_{-0.477}$ | $1.028^{+0.038}_{-0.049}$ | $12.840^{+0.061}_{-0.062}$ | $10.541^{+1.073}_{-1.048}$ | $0.528^{+0.335}_{-0.593}$ | $0.027^{+0.457}_{-0.420}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$11.565^{+0.039}_{-0.031}$ | $0.356^{+0.102}_{-0.114}$ | $0.842^{+0.047}_{-0.045}$ | $12.649^{+0.065}_{-0.070}$ | $12.316^{+0.121}_{-0.135}$ | $0.559^{+0.292}_{-0.349}$ | $-0.634^{+0.564}_{-0.203}$
Figure 6: Posterior 1$\sigma$ and 2$\sigma$ intervals of galaxy assembly bias
parameters. The two panels show the 1$\sigma$ (thick error bars) and 2$\sigma$
(thin error bars) constraints on the central and satellite assembly bias
parameters, $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$, respectively. The
statistic sets from which the constraints are derived are shown in different
colours ($n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$ in blue;
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$ in
orange). The different samples are laid out vertically and labelled on the
$y$-axis. The markers show the medians in the marginalised posterior
distributions, the thick error bars span between the 16th and 84th
percentiles, and the thin error bars span between the 2.5th and 97.5th
percentiles. The zero point, at which there is no assembly bias, is marked
with a gray dashed line.

###  5.2 Full Posterior of Model Parameters

In addition to the constraints listed in Table 4, we also depict the full
posterior of all model parameters visually in a series of figures. Figure 7
shows results for the $M_{r}<-21$ sample, Figure 8 shows results for the
$M_{r}<-20$ sample, and Figure 9 gives the $M_{r}<-19$ results. We do not show
two-dimensional constraints for $M_{r}<-20.5$ and $M_{r}<-19.5$ in the
interest of brevity, since they do not add any qualitatively new information.
The figures show the 1$\sigma$ and 2$\sigma$ marginalised constraints in two
dimensions for all pairwise combinations of parameters. In each of these
figures, constraints on the five parameters of the HOD are shown in the upper
right triangle, and constraints on the seven parameters of the dHOD are
exhibited in the lower right triangle. In each triangle, each diagonal panel
shows the entire one-dimensional marginalised posterior distribution for the
parameter listed for the column. Furthermore, each panel shows results from
fits to clustering only
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$”) as well as results
from the simultaneous fits to clustering and CIC distributions
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$+$P(N_{\mathrm{CIC}})$”),
in different colours.

There are several points to note from these figures. In each of the fits to
clustering (“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$”), there are
prominent degeneracies among the parameters. For example,
$\sigma_{\mathrm{log}M}$ and $\log M_{\mathrm{min}}$ are degenerate, as are
$\alpha$ and $\log M_{1}$. These degeneracies are well known and have been
discussed in the earlier literature (e.g. Zentner et al., 2019). As suggested
by our earlier analysis in W19, the addition of $P(N_{\mathrm{CIC}})$ to the
set of observables partly breaks these degeneracies, resulting in considerably
more restrictive parameter constraints. In many cases, the improvement is
dramatic. For example, the width of the marginalised 1$\sigma$ constraint on
$\log M_{\mathrm{min}}$ is reduced by a factor of $\sim 4$ for the HOD
analysis of the $M_{r}<-20$ sample and by a factor of nearly 13 for the dHOD
analysis of the same sample.

Interestingly, the $M_{r}<-19.0$ sample shows a bimodality in the inferred
posterior distribution of $A_{\mathrm{sat}}$ from the
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$
analysis. The effect of non-zero satellite assembly bias is to distribute more
satellite galaxies into a subpopulation of haloes, which enhances the small-
scale clustering regardless of the sign of $A_{\mathrm{sat}}$. In the jargon
of the HOD model, this is an enhancement of the “one-halo” clustering. This
small-scale, one-halo clustering is enhanced regardless of the sign of
$A_{\mathrm{sat}}$ (regardless of whether or not low-concentration or high-
concentration haloes preferentially host satellite galaxies) because placing a
fixed number of galaxies into fewer haloes increases the number of pairs among
these galaxies. Consequently, both $A_{\mathrm{sat}}>0$ and
$A_{\mathrm{sat}}<0$ yield enhanced small-scale clustering, and it is not at
all surprising to find that when values of $A_{\mathrm{sat}}\neq 0$ are
preferred by data, there is significant probability in the posterior
distribution corresponding to both $A_{\mathrm{sat}}<0$ and
$A_{\mathrm{sat}}>0$. The enhancement is not perfectly symmetric between
positive and negative values of $A_{\mathrm{sat}}$ for a variety of reasons
(Hearin et al., 2016). Moreover, this degeneracy is partially broken by the
addition of $P(N_{\mathrm{CIC}})$, which contains higher-order information of
the small-scale galaxy distribution, and is able to better distinguish between
positive, zero, and negative values of $A_{\mathrm{sat}}$.

Notice additionally, that there are some interesting notes of tension among
our analyses. In the HOD analysis of the $M_{r}<-20$ sample (Figure 8, upper
right), there is some weak tension between the clustering-only fits
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$”) and the fits that
include both clustering and CIC
(“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$+$P(N_{\mathrm{CIC}})$”).
While this tension is weak, it is also mitigated in the dHOD fit to the same
sample, which includes assembly bias (Figure 8, lower left). More interesting
in this regard may be the $M_{r}<-19$ sample. The HOD analysis of the
$M_{r}<-19$ sample exhibits more sizeable tensions (Figure 9, upper right).
Furthermore, this tension persists in the dHOD analysis which includes
assembly bias (Figure 9, lower left). We discuss this tension further in
Section 6.3.

Figure 7: The standard HOD and decorated HOD fits for the $M_{r}<-21.0$
sample. Fits for the standard HOD are shown in the upper right triangle, and
fits for the decorated HOD, which has two additional assembly bias parameter,
are shown in the lower left triangle. The contours show the $1\sigma$ and
$2\sigma$ constraints as two-dimensional projections in the parameter space,
where the $1\sigma$ contours are plotted in thicker lines. The one-dimensional
histograms are marginalised for each individual parameter. The two sets of
contours with different colours show the constraints from $n_{\mathrm{gal}}$
and $w_{\mathrm{p}}(r_{\mathrm{p}})$ (blue), and the constraints from
$n_{\mathrm{gal}}$, $w_{\mathrm{p}}(r_{\mathrm{p}})$ and $P(N_{\mathrm{CIC}})$
combined (orange), respectively. The discontinuities in the contours are
plotting artefacts and do not reflect irregularities in the constraints.
Figure 8: Same as Figure 7, but for the $M_{r}<-20.0$ sample. Figure 9: Same
as Figure 7, but for the $M_{r}<-19.0$ sample.

##  6 Interpretation

The initial motivation of this work was to leverage the degeneracy breaking
power of CIC statistics to complement the two-point function in order to
constrain and/or detect galaxy assembly bias. The results of the previous
section demonstrate that we have achieved this goal. Our current task is now
to interpret these results. In this section, we examine the degree to which
our results may be interpreted as a detection of assembly bias, the comparison
between the standard HOD and the decorated HOD results, and the trends among
different luminosity threshold samples.

###  6.1 Detection of Galaxy Assembly Bias

We begin with a discussion of the degree to which our results may be
interpreted as a detection of assembly bias in the context of the dHOD model.
In general, the data prefer positive $A_{\mathrm{cen}}$, indicating that
central galaxies of each luminosity threshold are more likely to reside within
high-concentration haloes. In several samples, $A_{\mathrm{cen}}\leq 0$ is
ruled out at relatively high significance. We cast the strength of these
assembly bias detections into approximate “sigma level measures,” using the
method described in Section 2.2 of Köhlinger et al. (2019). Namely, we derive
posterior distributions of the differences between the parameters in the
chains and parameter instances without assembly bias, quantify the rareness of
zero difference, and convert it to the sigma level according to the
probability distribution of a one-dimensional Gaussian. Strong levels of
positive central assembly bias are observed in the $M_{r}<-20.0$ (at least
4.4$\sigma)$ and $M_{r}<-19.5\ (3.4\sigma)$ samples. Positive central assembly
bias is also preferred by the $M_{r}<-20.5\ (2.2\sigma)$ and $M_{r}<-19.0\
(1.8\sigma)$ samples. We find no evidence of central galaxy assembly bias in
the brightest $M_{r}<-21.0$ sample, from which constraints are generally quite
weak due to the larger statistical errors on this sample. In comparison, we
find only relatively weak indications that $A_{\mathrm{sat}}$ may be negative.
Non-zero satellite assembly bias is only marginally suggested in the
$M_{r}<-20.0\ (1.0\sigma)$ and $M_{r}<-19.0\ (1.4\sigma)$ samples, and not
suggested by data from the other samples.

The senses of these signals are consistent with our expectations. At a fixed
halo mass, haloes with deeper potential wells and thus higher concentrations
are more likely to host central galaxies. On the other hand, haloes with
higher concentrations tend to have less surviving substructure, and therefore
fewer satellite galaxies. We discuss the physical origin of our signals in
more detail in Section 7.1, referring to predictions of hydrodynamical
simulations.

###  6.2 Information Criteria of HOD and dHOD Models

The preceding discussion presents the case that this analysis does represent a
detection of assembly bias. However, it is necessary to explore the degree to
which the data prefer the dHOD model to the standard HOD model, as the
decorated HOD (dHOD) is an extension to the standard HOD model.

We quantify our model selection using the Akaike Information Criterion (AIC,
Akaike, 1973) and the Bayesian Information Criterion (BIC, Schwarz, 1978),
both of which penalise models for having more parameters. The information
criteria are

$\displaystyle{\rm AIC}=2k-2\ln\mathcal{L}_{\rm max},$ (10) $\displaystyle{\rm
BIC}=k\ln n-2\ln\mathcal{L}_{\rm max},$ (11)

where $k$ is the number of parameters in the model, $n$ is the number of data
points involved in the fit, and $\mathcal{L}_{\rm max}$ the maximum
likelihood, in our case $-2\ln\mathcal{L}_{\rm max}=\chi^{2}_{\rm min}$. The
model with the smaller AIC or BIC value is considered a better description of
the data. Generally, an extended model is considered to be strongly preferred
if it reduces the AIC or BIC by 5 or more (e.g., Burnham et al., 2011).

The information criteria, along with the minimum $\chi^{2}$ values normalised
by the number of degrees of freedom, are listed in Table 5. Our choice not to
incorporate the Hartlap factor (see Section 4.4) may have a moderate effect on
the absolute values of $\chi^{2}$ and the information criteria, but does not
affect the comparison between models. Comparing between the rows of HOD and
dHOD fits for each sample, we find mixed results. The fits in which clustering
alone is used to constrain the galaxy-halo connection models
(“$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$”) do not prefer the dHOD
over HOD and even give a marginal preference to the standard HOD relative to
the dHOD.

Including CIC statistics in the analysis
(“$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$”
analyses) also yields mixed conclusions. In these analyses, the dHOD model is
often preferred. In fact, this preference is categorised as strong in the
$M_{r}<-20$ and $M_{r}<-19.5$ samples. However, the opposite is true in the
brightest sample. Our analysis of the $M_{r}<-21$ sample shows a preference
for the HOD over the dHOD. It may be the case that assembly bias is not
indicated in the most luminous sample because concentration-dependent halo
clustering weakens as halo mass increases. Therefore, a concentration-
dependent halo occupation model induces only small changes in predicted galaxy
clustering statistics in high-luminosity (i.e., high-mass) samples.

Table 5: Model selection and information criteria. For each galaxy sample, we list several measures for assessing the effectiveness of the HOD and dHOD models to fit the data measurements. The measures include the $\chi^{2}$ value normalised by the degree of freedom, the Akaike Information Criterion, and the Bayesian Information Criterion. These are calculated for both sets of statistics, $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$, and $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$. We also list the level of tension between the parameters inferred from the two statistic sets, as a proxy for how well the model can fit all the statistics at once. |  $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$ |  | $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$ |  | Tension
---|---|---|---|---|---
|  $\chi^{2}$/DoF | AIC | BIC |  |  $\chi^{2}$/DoF | AIC | BIC |
$M_{r}<-21.0$, HOD | 0.71 | 15.65 | 18.47 |  | 0.38 | 19.15 | 25.99 |  | 0.20$\sigma$
dHOD | 0.69 | 18.11 | 22.07 |  | 0.42 | 23.19 | 32.76 |  | 0.07$\sigma$
$M_{r}<-20.5$, HOD | 1.38 | 21.06 | 23.88 |  | 0.81 | 34.40 | 42.18 |  | 0.93$\sigma$
dHOD | 1.25 | 21.49 | 25.45 |  | 0.62 | 31.35 | 42.24 |  | 0.18$\sigma$
$M_{r}<-20.0$, HOD | 1.49 | 21.91 | 24.74 |  | 1.54 | 62.49 | 70.81 |  | 0.89$\sigma$
dHOD | 1.18 | 21.08 | 25.04 |  | 1.05 | 47.69 | 59.34 |  | 0.07$\sigma$
$M_{r}<-19.5$, HOD | 0.60 | 14.78 | 17.60 |  | 1.07 | 48.35 | 56.92 |  | 0.87$\sigma$
dHOD | 0.57 | 17.40 | 21.36 |  | 0.81 | 41.63 | 53.61 |  | 0.25$\sigma$
$M_{r}<-19.0$, HOD | 0.77 | 16.17 | 19.00 |  | 2.09 | 87.39 | 96.08 |  | 2.00$\sigma$
dHOD | 0.92 | 19.51 | 23.47 |  | 1.99 | 83.65 | 95.82 |  | 1.84$\sigma$

###  6.3 Tension Between Analyses

When constraining a model using different data sets, the results could be in
tension with each other. In other words, the model is unable to simultaneously
reproduce the different data, which could indicate incompleteness in the
modelling choices. The consistency between constraints from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$ and
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$ thus
provides a test of the model’s ability of describing the real Universe. We
quantify this consistency in Section 6.2, where we compare the performance of
the HOD and the dHOD models. A consequence of tensions between statistics is a
decrease in the constraining power of the joint analysis, as the model
struggles to fit both observables. This could explain the behaviour of the
$M_{r}<-19.0$ sample, a rare case for which the constraints on $\alpha$ and
$\log M_{1}$ worsen when $P(N_{\mathrm{CIC}})$ is included, as the preferences
for those parameters indeed show a significant discrepancy between the data
sets.

Also in Table 5, we quantify the consistency between the two sets of parameter
constraints, with and without $P(N_{\mathrm{CIC}})$, again using the method in
Köhlinger et al. (2019). As discussed previously, this measure indirectly
reflects the ability of the models to describe the underlying physics. In most
of the samples, we find no significant tension between the data sets for
either model, however, we note that the dHOD model shows a universally
superior consistency between data sets. For the $M_{r}<-19.0$ sample, there is
a 2.0$\sigma$ tension between the observable sets, which is slightly
alleviated (1.8$\sigma$) when including assembly bias. The tension could be
due to specific modelling choices, such as the simple form of the spatial
profile and velocities of satellites in haloes, especially as $M_{r}<-19.0$ is
the faintest sample and has the highest fraction of satellite galaxies. It
could also be due to the form of incorporation of galaxy assembly bias, or the
choice to only categorise haloes into two discrete subpopulations by
concentration. Overall, the decorated HOD is a superior model for accounting
for the different statistics at once, though there is still room for
improvement in the modelling choices.

###  6.4 Comparison Across Galaxy Samples

The different galaxy samples represent populations that have different
luminosities, reside in different environments, and are observed in different
quantities. The model parameters, as well as the constraining power, show
dependencies on the luminosity threshold. Our conclusions regarding galaxy
assembly bias are also different among the different samples. We discuss the
trends with luminosity thresholds in this subsection.

We first examine how parameter values vary with the luminosity threshold used
for sample selection. The three mass scale parameters, $\log
M_{\mathrm{min}}$, $\log M_{1}$ and $\log M_{0}$, show a clear trend of
increase when going to brighter samples. This is consistent with our
expectation that brighter galaxies live in more massive haloes. Overall
$\sigma_{\mathrm{log}M}$ becomes smaller for brighter samples, which suggests
that the scatter in the stellar mass–halo mass relation tends to be smaller on
the more massive end (in agreement with Lange et al., 2019a), but the trend is
not very pronounced, and there are exceptions. There isn’t a clear and
consistent trend in $\alpha$ with luminosity.

Another comparison can be made between the tightness of constraints for the
different luminosity samples, which is not a monotonic function of luminosity.
Instead, the constraining power is lower on both the faint and the bright
ends, and reaches a maximum for the intermediate samples. This is generally
true for the different parameters, and consistent with our expectations. The
brighter samples have lower number densities, and the errors of the statistics
are dominated by shot noise, whereas the fainter samples are limited by flux,
in which case cosmic variance due to finite volume becomes the main source of
uncertainty. The intermediate samples are not as severely susceptible to
either effect as the more extreme samples, and the balance leads to the best
constraints that we can obtain. We also note the difference between the impact
of small-scale shot noise on $w_{\mathrm{p}}(r_{\mathrm{p}})$ and
$P(N_{\mathrm{CIC}})$ in the bright samples. The constraining power of
$w_{\mathrm{p}}(r_{\mathrm{p}})$ comes both from smaller and larger radii,
while $P(N_{\mathrm{CIC}})$ focuses on the higher-order clustering at small
scales, which causes $P(N_{\mathrm{CIC}})$ to be even more sensitive to the
decrease of galaxy numbers, and undermines its ability to put constraints on
parameters. This is indeed what we observe on the bright end of the samples:
constraints from
$n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$ worsen
much more than $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$ only.

An additional factor contributes to the luminosity dependence of the
constraints on the assembly bias component of the model. The manifestation of
galaxy assembly bias in galaxy clustering relies on the existence of halo
assembly bias, which is the dependence of halo clustering on secondary halo
properties. Strong levels of halo assembly bias therefore amplify the effect
of galaxy assembly bias on galaxy clustering, allowing easier detection. In
the halo mass range relevant to our galaxy samples, the dependence of halo
clustering strength on halo concentration weakens toward the high mass end
(Wechsler et al., 2006; Gao & White, 2007), and further suppresses any galaxy
assembly bias signal that may be present in the $M_{r}<-21.0$ sample.

##  7 Discussion

In this section, we look into the physics behind our results, discuss our
modelling choices, and compare our findings with previous work.

###  7.1 Galaxy Assembly Bias in Hydrodynamical Simulations

The results presented in the previous sections strongly suggest the existence
of central galaxy assembly bias in the sense that central galaxies of a fixed
luminosity are more likely to inhabit high-concentration haloes at fixed halo
mass. The results also provide a hint of satellite galaxy assembly bias, where
galaxies of a fixed luminosity are more likely to inhabit low-concentration
halos. The senses of these assembly bias signals are in line with our gross
physical intuition. Central galaxies are more likely to form in haloes with
deeper potential wells at fixed mass. These haloes correspond precisely to
high-concentration haloes. Moreover, at a given halo mass, haloes that
assembled earlier have higher concentrations (e.g., Navarro et al., 1997) and
less surviving substructure that may host satellite galaxies (e.g., Zentner et
al., 2005; Mao et al., 2015; Jiang & van den Bosch, 2017). Correspondingly,
low-concentration haloes tend to have more satellite haloes at fixed mass.
Hence, both a positive $A_{\mathrm{cen}}$ and a negative $A_{\mathrm{sat}}$
are in line with our physical expectations.

There have been a number of studies of assembly bias within galaxy formation
simulations. The most recent of these is Yuan et al. (2022), who studied
assembly bias in galaxy samples selected from the IllustrisTNG-300-1
simulation to mock the DESI LRG and ELG galaxy samples. While neither of these
samples is directly comparable to the SDSS DR7 Main Galaxy Sample, it is
interesting to note that the LRG sample of galaxies exhibits galaxy assembly
bias in the same sense as what we observe in our study. In particular, central
galaxy LRGs in IllustrisTNG preferentially inhabit haloes with concentrations
that are higher than average while satellite LRGs preferentially inhabit
haloes with lower concentrations. On the other hand, both central and
satellite ELGs are found preferentially within haloes with concentrations that
are lower than average, a result that is not surprising because ELGs are
thought to be preferentially newly forming galaxies that are found
preferentially within newly forming haloes.

A number of other studies are worthy of note. Montero-Dorta et al. (2021)
studied assembly bias more generally within IllustrisTNG-300-1. Broadly
speaking, these authors found that galaxies with stellar masses above
$10^{7}\,\mathrm{M}_{\odot}$ preferentially inhabit early-forming haloes and
as early-forming haloes are preferentially high-concentration haloes, this
result is qualitatively consistent with the enhanced probability of central
galaxies to reside in high-concentration host haloes at fixed mass which we
infer from the SDSS data. Hadzhiyska et al. (2021) compared assembly bias
present in the IllustrisTNG-300-1 simulation with that in the Santa Cruz semi-
analytic galaxy formation model (Gabrielpillai et al., 2021). In both the
simulation and the semi-analytic model, central galaxies preferentially reside
within haloes of higher concentration at fixed halo mass while satellite
galaxies are found with higher probability in lower-concentration haloes. This
is consistent with the earlier findings of Bose et al. (2019) who studied
galaxy clustering in IllustrisTNG-100-1 and IllustrisTNG-300-1 simulations. Xu
& Zheng (2020) came to similar conclusions in their study of the Illustris
galaxies and determined that most of the effect of assembly bias could be
accounted for by tabulating the galaxy content of haloes as a function of peak
circular velocity, rather than halo mass. Contreras et al. (2019) studied
assembly bias in the semi-analytic galaxy formation model of Guo et al.
(2013), finding results in qualitative agreement with the aforementioned
studies. Each of these recent studies are consistent with earlier work on the
EAGLE simulation (Chaves-Montero et al., 2016). Interestingly, all of these
results from studies of simulations and semi-analytic models are in
qualitative agreement with the sense of assembly bias that we infer from our
SDSS analyses.

###  7.2 Modelling of Galaxy Assembly Bias

In this work, we obtain constraints on galaxy assembly bias in terms of the
decorated HOD model, with halo concentration as the secondary halo property.
We make this choice for several reasons.

Of the commonly considered halo properties, halo concentration has been found
to exhibit strong correlations with both halo assembly history (e.g., Wechsler
et al., 2002) and halo clustering (e.g., Wechsler et al., 2006). The former is
closely connected with galaxy formation, while the latter is a potential
source of observable signals, rendering concentration likely to yield a
detection of galaxy assembly bias. Meanwhile, other secondary halo properties
do not simultaneously show strong levels of correlation with galaxy properties
and halo clustering. For example, halo spin, which is also strongly correlated
with halo clustering (e.g., Mao et al., 2018), was recently found to be
ineffective in predicting galaxy spin and size (Jiang et al., 2019), contrary
to previous belief. Our foremost objective being to determine whether any form
of galaxy assembly bias exists, concentration is a favourable choice. It may
be interesting to investigate galaxy assembly bias signals with respect to
other halo properties, that are intrinsically weaker, when larger data volumes
become available.

There are several galaxy–halo connection models, besides the decorated HOD,
that incorporate adjustable galaxy assembly bias, e.g., the interpolated
abundance matching model in Lehmann et al. (2017), the extended HOD model in
Salcedo et al. (2020), etc. The advantage of the decorated HOD is that it
treats central and satellite galaxies separately and the values of the
assembly bias parameters are straightforward to interpret. We have shown that
a mass-only model such as the HOD, though relatively flexible and successful
in reproducing two-point clustering behaviour, struggles to explain our
combined statistics in certain galaxy samples. We therefore tentatively expect
that signals of galaxy assembly bias would also be evident in constraints on
alternative models, and we leave a detailed investigation to future work.

###  7.3 Comparison to Other Work

In relation to our primary results regarding assembly bias, there are few
previous observational works with which to compare. Zentner et al. (2019) and
Vakili & Hahn (2019) studied two-point clustering constraints on assembly bias
using a forward-modelling approach. Our results are in broad general agreement
with these previous studies. Both Zentner et al. (2019) and Vakili & Hahn
(2019) observed marginal indications of central galaxy assembly bias in the
$M_{r}<-19.5$, $M_{r}<-20$, and $M_{r}<-20.5$ threshold samples of the same
sense as our detections. Our constraints on $A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$ are typically substantially more restrictive than the
constraints of Zentner et al. (2019) or Vakili & Hahn (2019). Taken at face
value, our results indicate that $A_{\mathrm{cen}}>0$ (positive assembly bias)
for the $M_{r}<-20$ sample with at least $4.4\sigma$ significance.

A number of additional authors have studied the clustering of the SDSS DR7
sample using an HOD approach, without studying assembly bias in those samples.
These include Zehavi et al. (2011), Guo et al. (2015) (who studied redshift-
space clustering), and Szewciw et al. (2022) (who studied projected
clustering, redshift-space clustering, group multiplicity, and several other
observables as part of a campaign to constrain the standard HOD optimally).
Our constraints on the standard HOD parameters are broadly consistent with the
constraints from all of these studies. The precision with which we constrain
the HOD parameters is generally better than that from Zehavi et al. (2011) and
Guo et al. (2015) and lies between the “NWG” and “OPT” analyses in Szewciw et
al. (2022), with the “OPT” constraints often being a factor of 2-3 more
restrictive than ours. As we were completing this work, we became aware of an
independent study that analyses SDSS DR7 samples, building upon the framework
of Szewciw et al. (2022), which reaches broadly similar conclusions (Beltz-
Mohrmann et al., 2022). In particular, they also find evidence for positive
central assembly bias in the samples they study.

While our clustering only,
“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$” analyses are broadly
consistent with the result of previous studies of the HOD, the introduction of
cylinder counts does result in interesting deviations. To be specific,
including $P(N_{\mathrm{CIC}})$ in the analysis drives the preferred value of
$\alpha$ lower, often substantially lower than unity and substantially lower
than the values quoted in previous work. This trend is evident in our own work
in Table 4, where the
“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$+$P(N_{\mathrm{CIC}})$”
analyses yield inferred values of $\alpha$ substantially lower than the
clustering-only analyses. We suspect that this is related to the tension
introduced by cluster counts and discussed above, but we relegate a full study
of this shift to follow-up work.

##  8 Conclusions

We have reanalysed SDSS DR7 galaxy clustering data using a novel combination
of observables and with the aim of detecting and/or constraining galaxy
assembly bias. We used the observable combination of galaxy number density,
projected two-point clustering, and counts-in-cylinders, and provided new
measurements of each of these statistics. We interpreted these data by fitting
them with HOD and dHOD models of the galaxy–halo connection. The HOD formalism
is a standard "halo mass-only" model which assumes that galaxies occupy haloes
in a manner that depends only upon halo mass. Our particular implementation of
the dHOD included galaxy assembly bias by allowing central and satellite
occupations to vary with host halo concentration at a fixed halo mass. We
obtained updated constraints on both the HOD and dHOD models. In particular,
we constrained the level of galaxy assembly bias, as parametrised in the dHOD
model, more precisely than any previous studies.

Our primary conclusions are as follows.

  1. 1.

By adding the counts-in-cylinders statistic to the canonical analysis of
number density and projected two-point function, we achieve a considerable,
general improvement in the constraints on the galaxy–halo connection.

  2. 2.

We detect strong positive central galaxy assembly bias in the $M_{r}<-20.0$
and $M_{r}<-19.5$ samples, and relatively strong levels of positive central
assembly bias in the $M_{r}<-20.5$ and $M_{r}<-19.0$ samples, but find no
evidence of central assembly bias in the $M_{r}<-21.0$ sample.

  3. 3.

Negative satellite assembly bias is suggested in the $M_{r}<-20.0$ and
$M_{r}<-19.0$ samples, but we do not detect satellite assembly bias in the
other samples.

  4. 4.

The decorated HOD, which incorporates galaxy assembly bias, in most cases
provides a better description of galaxy two-point clustering and counts-in-
cylinders simultaneously than the standard mass-only HOD.

The implications of our findings are twofold. First, we have demonstrated the
constraining power of galaxy count statistics. There are two possible avenues
for acquiring additional observational information, in order to resolve
pending theoretical issues: either to use larger data sets and reduce error,
or to exploit existing data using more efficient statistical probes. The
former, though very promising, especially given the great effort that is going
into a new generation of galaxy surveys, does not solve the problem of
degeneracies in the model parameter space, for they are intrinsic to the
selected statistics, and determined by the manner of their dependence on the
model. The alternative approach, which is to develop new statistics, can be
more effective in this respect. In our case, count statistics incorporate the
higher-order information of the field, which breaks multiple degeneracies with
the two-point function, leading to tighter constraints in all dimensions. We
advocate the use of these statistics in combination with the commonly used
two-point measurements, which has the potential not only to improve
constraints on the galaxy–halo connection, but also on cosmological models and
other aspects of galaxy physics.

Second, based on the dHOD formalism, we have detected galaxy assembly bias in
several galaxy samples, which indicates that the first-order assumption that
galaxy occupation only depends on halo mass is incomplete. As was shown in Wu
et al. (2008); Zentner et al. (2014) and McCarthy et al. (2019), neglecting
galaxy assembly bias when it is present in the real Universe may lead to
systematic errors in cosmological inferences. Our detection of galaxy assembly
bias further necessitates explorations of its physical origin and better
models that incorporate the effect. Traditional models that adopt the mass-
only ansatz will not suffice in the new era of precision cosmology, and
developing new models that account for galaxy assembly bias is a timely task,
which is attracting increasing attention (see, for example, Yuan et al., 2021;
Contreras et al., 2021). In addition, the nature of galaxy assembly bias can
be studied from several perspectives, including its dependence on cosmic time,
its roots in halo assembly bias, the galaxy and halo properties that best
capture the correlation, and the relevant physical processes in galaxy
formation and evolution (see, e.g., Lacerna et al., 2014; Chaves-Montero et
al., 2016; Artale et al., 2018; Contreras et al., 2019; Montero-Dorta et al.,
2020). We aim to explore other applications of galaxy count statistics, and
further look into the physical origin of galaxy assembly bias in future work.

## Acknowledgements

The authors thank Dhayaa Anbajagane, Camille Avestruz, Michael Blanton,
Phillip Mansfield, Li-Cheng Tsai, Sihan Yuan, and Rongpu Zhou for useful
discussions.

KW acknowledges support from the Leinweber Postdoctoral Research Fellowship at
the University of Michigan, DoE Award DE-SC009193, and the Mellon Fellowship
at the University of Pittsburgh. YYM was supported by NASA through the NASA
Hubble Fellowship grant no. HST-HF2-51441.001 awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-26555. KW and
ARZ were partially supported by a grant from the US National Science
Foundation (NSF) through grant AST 1517563. KW, LM, and ARZ were partially
supported by the Pittsburgh Particle Physics Astrophysics and Cosmology Center
(PITT PACC) and the Department of Physics and Astronomy at the University of
Pittsburgh. HG is supported by National Science Foundation of China (Nos.
11922305, 11833005, 12011530159). FvdB is supported by the National
Aeronautics and Space Administration through Grant No. 19-ATP19-0059 issued as
part of the Astrophysics Theory Program.

This research made use of Python, along with many community-developed or
maintained software packages, including IPython (Perez & Granger, 2007),
Jupyter (jupyter.org), Matplotlib (Hunter, 2007), NumPy (van der Walt et al.,
2011), SciPy (Jones et al., 2001), Astropy (Astropy Collaboration et al.,
2013), Pandas (McKinney, 2010), Corrfunc (Sinha, 2016) and Mangle (Hamilton &
Tegmark, 2004; Swanson et al., 2008)444We use an implementation in Python from
https://github.com/esheldon/pymangle.. This research made use of NASA’s
Astrophysics Data System for bibliographic information.

## Data Availability

The data underlying this article were accessed from
http://sdss.physics.nyu.edu and https://www.cosmosim.org. The derived data
generated in this research are available in the article, and any additional
data will be shared on reasonable request to the corresponding author.

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##  Appendix A Validation of Statistic Estimation

Figure A.1: Comparison between the cube mock and the cone mocks of the galaxy
number density $n_{\mathrm{gal}}$. Each panel compares a different luminosity
sample. The underlying cube value is shown as a vertical black line. The
horizontal black error bars are the total jackknife error from the cube and
the cones. The values measured from the different light cone mocks are plotted
as a histogram, and the mean is marked by the vertical red line. This figure
shows that the measurement of the galaxy number density is consistent between
our cube and cone mocks for all five samples. Figure A.2: Comparison between
the cube mock and the cone mocks of the projected two-point function
$w_{\mathrm{p}}$. The five panels show the fractional deviation between the
cube and the cones for the different luminosity samples, labelled at the top.
The horizontal black line marks zero deviation, and the error bars show the
total jackknife error from the cube and the cones. Individual cones are
plotted as thin lines in the background, and the mean of the cones is shown by
the solid red line. Measurements for all the samples are consistent within
error. Figure A.3: Comparison between the cube mock and the cone mocks of the
counts-in-cylinders statistic $P(N_{\mathrm{CIC}})$. Same as Figure A.2, but
for counts-in-cylinders instead of $w_{\mathrm{p}}$. Note that the range of
the $x$-axis is different for each sample, though $y$-axis ranges are the same
for all. Measurements are again consistent for all the samples.

In order to fit SDSS data using a simulation, we need measurements of the
observable statistics from the simulation to reflect the behaviour of real
data in an unbiased manner. However, light cone mocks that mimic observed data
are computationally expensive, and therefore infeasible for sampling from the
high dimensional model parameter space, whereas measurements from the original
cubic volume of the simulation can be obtained much more rapidly. Consistency
between statistic measurements from the cube and the light cone mocks that
have the same underlying physics would validate the use of the cube for
fitting data.

In this appendix, we detail the construction of cone mocks that incorporate
the relevant geometrical and observational effects present in SDSS data,
obtain measurements on the cone mocks with the exact same algorithm used for
SDSS data, and demonstrate that our treatment of observational effects ensures
that cube mocks yield unbiased estimates of the statistics measured in cone
mocks, and hence observational data.

###  A.1 Halo Populating in Simulation Cube

We populate the halo catalogue at $z=0.1$ from SMDPL using the stellar
mass–(sub)halo mass relation model described in Behroozi et al. (2010) to get
a galaxy catalogue in the cubic simulation volume (hereafter the “cube mock”).
This model is implemented in the halotools software package (Hearin et al.,
2017). The model provides the position, velocity, and stellar mass information
of the galaxies. We assume a constant mass-to-light ratio for all galaxies,
such that the ranking of galaxy stellar masses from large to small is
equivalent to the ranking of absolute magnitudes from bright to faint.

Since we are only interested in studying the consistency of galaxy statistics
between differently constructed mocks, the specific choice of any reasonable
galaxy–halo connection model in this step should not affect our results.

###  A.2 Cone Building

We build cone mocks (hereafter “cone mocks”) from the cube mock described in
Appendix A.1, with a routine that is similar to Lange et al. (2019a). Note
that we do not populate the halo catalogue again, but directly use the cubic
galaxy catalogue, such that the cube mock and cone mocks have identical
underlying cosmologies and galaxy–halo connections. The steps are as follows:

  * •

Choose a random position in the cubic volume to place our virtual observer;

  * •

Periodically repeat the cube mock out to the desired depth;

  * •

Calculate and record the true redshifts $z_{\rm true}$ of the galaxies,
accounting for both distance and velocity information;

  * •

Apply redshift measurement uncertainty according to the model described in
Appendix A of Guo et al. (2015), and record the resulting redshifts with
error, $z_{\rm true,err}$;

  * •

Choose a random direction of observation, convert the galaxy positions into
angular coordinates $\rm{(ra,dec)}$, and apply the 7461 $\rm{deg}^{2}$-SDSS
footprint;

  * •

Again assuming a constant mass-to-light ratio for all galaxies, rank the
$r$-band apparent magnitudes of galaxies

  * •

Keep galaxies with $0.01<z_{\rm true,err}<0.18$ and make a flux limit
selection based on the apparent magnitude ranking and the total number of
galaxies in the same redshift range in SDSS;

  * •

Assign fibre collision status to galaxies, using the method described in Guo
et al. (2012), which takes into account the tiling scheme of SDSS, and
includes random collisions with background galaxies and other failures, in
order to match the observed fractions of fibre collided galaxies;

  * •

Perform the nearest-neighbour correction for the fibre collision effect, as
was done in real data, and record the resulting $z_{\rm obs}$.

We repeat this process with different random seeds to generate 100 cone mocks.

In summary, the main distinctions between the cube and cone mocks include: (i)
the cone mocks mimic the angular coordinate system of observational data,
whereas the cube mock adopts the plane-parallel approximation; (ii) the cone
mocks account for the observational flux limit; (iii) the cone mocks have the
geometry of the SDSS footprint; and (iv) the cone mocks incorporate the fibre
collision effect.

###  A.3 Sample Selection

For validation purposes, we define samples in the mock catalogues by galaxy
number density, that correspond to the five data samples in the first part of
Table 1. This ensures similar discreteness noise in the mocks as in the real
data for each sample.

For the cube mock, we multiply the number densities listed in Table 2 by the
simulation volume to get the total number of galaxies $N_{\rm cube}$ in each
sample. We then select the $N_{\rm cube}$ brightest galaxies from the cube
mock according to the magnitude ranking, which we identify as the luminosity-
threshold samples. For the cone mocks, in addition to the same magnitude
thresholds as in the cube, we apply the same redshift range limits on $z_{\rm
obs}$ as those used for selecting data samples555The galaxy luminosity
function in our mock slightly differs from that in the survey, which
propagates to the effective flux limit in our galaxy selection. To avoid
exceeding the mock flux limit, we adopt a conservative redshift cut for the
$M_{r}<-19.5$ threshold, at $cz_{\rm max}=23450\ \mathrm{km\ s^{-1}}$ in the
mocks instead of the $25450\ \mathrm{km\ s^{-1}}$ limit for data, which does
not affect our results qualitatively..

###  A.4 Cube Mock Algorithm

####  A.4.1 Statistics

In cube mocks, we apply periodic conditions and adopt the plane-parallel
approximation. We place the line of sight along the three axes in turn, apply
redshift space distortions from galaxy peculiar velocities for each
respectively, and average the resulting measurements for each observable.

By construction, the number density $n_{\mathrm{gal}}$ of each mock sample is
simply that of the corresponding data sample. For cube measurements of
$w_{\mathrm{p}}(r_{\mathrm{p}})$, we use the natural estimator for
$\xi(r_{\mathrm{p}},\pi)$,

$\hat{\xi}_{\rm N}=\frac{\rm{DD}}{\rm{RR}}-1,$ (12)

where DD is the normalised galaxy–galaxy pair count, and RR is the normalised
random–random pair count. Given the simplicity of the geometry, we use
analytic randoms instead of actually drawing random points to reduce the
computational cost. Galaxy pairs are counted in $r_{\mathrm{p}}$ bins and
$\xi(r_{\mathrm{p}},\pi)$ is integrated along the chosen axis out to
$\pi_{\mathrm{max}}$.

To measure $P(N_{\mathrm{CIC}})$, we centre a cylinder on every galaxy in the
sample, and count the number of companion galaxies that fall in the cylinder,
excluding the cylinder centre itself. The histogram of the counts is
calculated with our specified bins.

####  A.4.2 Jackknife Covariance

To test the consistency between cube and cone measurements, we need to
understand the uncertainty of both. Because the cube and cone mocks have the
same galaxy population, the only component of the covariance is the jackknife
covariance, which provides an estimate of the fluctuation among different
parts of the simulation volume. The division of the cube into jackknife cells
is trivial. We divide the simulation volume into 100 cuboids of $40\times
40\times 400(h^{-1}\mathrm{Mpc})^{3}$, where the long axis is the same length
as the simulation, and lies along the line of sight. We repeat the process for
each of the three projections and take the average jackknife covariance. The
total covariance matrix that we use for the consistency test is the sum of the
cube jackknife covariance and the cone jackknife covariance, where the cone
jackknife is estimated using the scaling with volume, $\mathbf{C}_{\rm
cone}=(V_{\rm cube}/V_{\rm cone})\mathbf{C}_{\rm cube}$ (see, for example,
Appendix A of Zheng & Guo, 2016).

###  A.5 Comparison Between Cube and Cone Mocks

We compare the measurements of $n_{\mathrm{gal}}$ (Figure A.1),
$w_{\mathrm{p}}(r_{\mathrm{p}})$ (Figure A.2), and $P(N_{\mathrm{CIC}})$
(Figure A.3) between the cube mock and the mean of the cone mocks generated
from it. We consider the measurements consistent between the cube and the
cones if the deviation is within the total jackknife error. In the figures, we
show the cube measurements with error bars, individual cone measurements, and
their mean. We find that for all the luminosity samples we consider, and for
all three of our statistics, the measurements are consistent within error.
This confirms that with our algorithm, measurements from the simulation cube
can be used as unbiased estimates of statistics measured from SDSS-like
datasets. In particular, we note that the nearest neighbour correction is
sufficient to account for fibre collision, for the statistics that we
consider.

##  Appendix B Results for Alternative Galaxy Samples

Table B.1: Inferred constraints on HOD and dHOD parameters for alternative galaxy samples. Same as Table 4, but for the alternative samples that exclude the Sloan Great Wall (see Table 1). $M_{r}<-20.0^{*}$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
---|---|---|---|---|---|---|---
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$12.081^{+0.323}_{-0.110}$ | $0.487^{+0.440}_{-0.315}$ | $1.099^{+0.048}_{-0.061}$ | $13.295^{+0.038}_{-0.056}$ | $10.744^{+1.172}_{-1.184}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$12.164^{+0.076}_{-0.105}$ | $0.578^{+0.098}_{-0.128}$ | $1.034^{+0.032}_{-0.050}$ | $13.294^{+0.033}_{-0.064}$ | $11.084^{+1.188}_{-1.425}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$12.150^{+0.430}_{-0.171}$ | $0.609^{+0.502}_{-0.395}$ | $1.061^{+0.060}_{-0.077}$ | $13.291^{+0.052}_{-0.077}$ | $10.846^{+1.147}_{-1.246}$ | $0.549^{+0.330}_{-0.652}$ | $-0.060^{+0.472}_{-0.436}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$12.105^{+0.110}_{-0.107}$ | $0.511^{+0.136}_{-0.166}$ | $0.955^{+0.076}_{-0.081}$ | $13.288^{+0.051}_{-0.096}$ | $11.899^{+0.592}_{-1.773}$ | $0.492^{+0.329}_{-0.441}$ | $-0.080^{+0.621}_{-0.740}$
$M_{r}<-19.5^{*}$ | $\log M_{\mathrm{min}}$ | $\sigma_{\mathrm{log}M}$ | $\alpha$ | $\log M_{1}$ | $\log M_{0}$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
HOD
$11.918^{+0.417}_{-0.175}$ | $0.623^{+0.519}_{-0.409}$ | $1.072^{+0.046}_{-0.059}$ | $13.047^{+0.045}_{-0.066}$ | $10.821^{+1.170}_{-1.240}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
HOD
$11.731^{+0.042}_{-0.031}$ | $0.295^{+0.117}_{-0.136}$ | $0.868^{+0.076}_{-0.077}$ | $12.784^{+0.115}_{-0.128}$ | $12.654^{+0.139}_{-0.196}$ | – – | – –
| $n_{\mathrm{gal}}+w_{\mathrm{p}}$
---
dHOD
$11.981^{+0.431}_{-0.233}$ | $0.723^{+0.512}_{-0.474}$ | $1.041^{+0.057}_{-0.075}$ | $13.043^{+0.063}_{-0.086}$ | $10.918^{+1.146}_{-1.301}$ | $0.506^{+0.352}_{-0.626}$ | $-0.066^{+0.492}_{-0.445}$
| $n_{\mathrm{gal}}+w_{\mathrm{p}}+P(N_{\mathrm{CIC}})$
---
dHOD
$11.717^{+0.034}_{-0.024}$ | $0.254^{+0.114}_{-0.136}$ | $0.808^{+0.071}_{-0.075}$ | $12.760^{+0.107}_{-0.124}$ | $12.671^{+0.143}_{-0.167}$ | $0.307^{+0.430}_{-0.538}$ | $-0.237^{+0.574}_{-0.541}$
Table B.2: Model selection and information criteria for alternative galaxy samples. Same as Table 5, but for the alternative samples that exclude the Sloan Great Wall. |  $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})$ |  | $n_{\mathrm{gal}}+w_{\mathrm{p}}(r_{\mathrm{p}})+P(N_{\mathrm{CIC}})$ |  | Tension
---|---|---|---|---|---
|  $\chi^{2}$/DoF | AIC | BIC |  |  $\chi^{2}$/DoF | AIC | BIC |
$M_{r}<-20.0^{*}$, HOD | 0.99 | 17.92 | 20.75 |  | 1.26 | 50.21 | 58.27 |  | 0.31$\sigma$
dHOD | 1.09 | 20.55 | 24.51 |  | 1.32 | 53.45 | 64.73 |  | 0.03$\sigma$
$M_{r}<-19.5^{*}$, HOD | 1.23 | 19.86 | 22.68 |  | 1.13 | 50.56 | 59.13 |  | 1.33$\sigma$
dHOD | 1.50 | 23.03 | 26.98 |  | 1.17 | 53.77 | 65.77 |  | 1.25$\sigma$

In this appendix, we list the fitting results for the alternatively selected
galaxy samples for the $M_{r}<-20.0$ and $M_{r}<-19.5$ luminosity thresholds,
which are described in Table 1. The alternative samples exclude the Sloan
Great Wall by adopting lower depth limits, and therefore have smaller volumes
and less constraining power.

The inferred HOD and dHOD parameters are listed in Table B.1, with the same
conventions as in Table 4. The model performance criteria are presented in
Table B.2, which is similar to Table 5.

For the alternative samples, we do not find significance evidence for non-zero
galaxy assembly bias. The only case where a positive $A_{\mathrm{cen}}$ is
marginally suggested is the
“$n_{\mathrm{gal}}$+$w_{\mathrm{p}}(r_{\mathrm{p}})$+$P(N_{\mathrm{CIC}})$”
analysis for the alternative $M_{R}<-20.0$ sample. We obtain weaker
constraints on the parameter space in general, relative to the constraints
from the main samples. It is unclear to what extent these differences are due
to the decrease in sample size or the exclusion of the Sloan Great Wall.