# Model BOSS & eBOSS Luminous Red Galaxies at $0.2<z<1.0$ using SubHalo
Abundance Matching with 3 parameters

Jiaxi Yu,1 Cheng Zhao,1 Chia-Hsun Chuang,2,3 Julian E. Bautista, 4 Ginevra
Favole,1 Jean-Paul Kneib,1,5 Faizan G. Mohammad, 6 Ashley J. Ross, 7 Anand
Raichoor, 8 Charling Tao, 4 Kyle Dawson, 2 Graziano Rossi 9
1Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL),
Observatoire de Sauverny, CH-1290 Versoix, Switzerland
2Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA
3Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,
452 Lomita Mall, Stanford, CA 94305, USA
4Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
5Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de
Marseille) UMR 7326, F13388, Marseille, France
6Waterloo Centre for Astrophysics, Dept. of Physics and Astronomy, University
of Waterloo, 200 University Ave. W., Waterloo ON N2L 3G1, Canada
7Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA
8Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA
9Department of Astronomy and Space Science, Sejong University, 209, Neungdong-
ro, Gwangjin-gu, Seoul, South Korea
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

SubHalo Abundance Matching (SHAM) is an empirical method for constructing
galaxy catalogues based on high-resolution $N$-body simulations. We apply SHAM
on the UNIT simulation to simulate SDSS BOSS/eBOSS Luminous Red Galaxies
(LRGs) within a wide redshift range of $0.2<z<1.0$. Besides the typical SHAM
scatter parameter $\sigma$, we include $v_{\rm smear}$ and $V_{\rm ceil}$ to
take into account the redshift uncertainty and the galaxy incompleteness
respectively. These two additional parameters are critical for reproducing the
observed 2PCF multipoles on 5–25$\,h^{-1}\,{\rm Mpc}$. The redshift
uncertainties obtained from the best-fitting $v_{\rm smear}$ agree with those
measured from repeat observations for all SDSS LRGs except for the LOWZ
sample. We explore several potential systematics but none of them can explain
the discrepancy found in LOWZ. Our explanation is that the LOWZ galaxies might
contain another type of galaxies which needs to be treated differently. The
evolution of the measured $\sigma$ and $V_{\rm ceil}$ also reveals that the
incompleteness of eBOSS galaxies decreases with the redshift. This is the
consequence of the magnitude lower limit applied in eBOSS LRG target
selection. Our SHAM also set upper limits for the intrinsic scatter of the
galaxy–halo relation given a complete galaxy sample: $\sigma_{\rm int}<0.31$
for LOWZ at $0.2<z<0.33$, $\sigma_{\rm int}<0.36$ for LOWZ at $0.33<z<0.43$,
and $\sigma_{\rm int}<0.46$ for CMASS at $0.43<z<0.51$. The projected 2PCFs of
our SHAM galaxies also agree with the observational ones on the 2PCF fitting
range.

###### keywords:

method: numerical – method: observational – galaxy: halo – cosmology: large-
scale structure of Universe

††pubyear: ††pagerange: Model BOSS & eBOSS Luminous Red Galaxies at
$0.2<z<1.0$ using SubHalo Abundance Matching with 3 parameters–25

##  1 Introduction

Lambda-Cold-Dark-Matter ($\Lambda$CDM) has been the standard cosmological
model since the 1990s. Under this framework, observations confirm that dark
matter (DM) is the dominant matter component of the Universe (e.g., Planck
Collaboration et al., 2020). DM particles interact only through gravity and
their gravitational evolution is assumed to start from a primordial Gaussian
random field with perturbations. If the over-density arising from the
perturbation on small scales is large enough, the infall of matter can
overcome the expansion of Universe, and form gravitational-bound clusters,
i.e., DM haloes. Perturbations on large scales evolve into web-like structures
known as the cosmic web (Bond et al., 1996). Baryonic matter may be captured
by haloes with deep potential wells and further evolve into galaxies (Press &
Schechter, 1974; White & Rees, 1978), a good tracer of the invisible cosmic
web.

Galaxy surveys explore the large-scale structure of the Universe by measuring
the redshifts of millions of galaxies and quasars. Their positions can be used
to calculate the two-point correlation function (2PCF) that encodes the
history of the universe expansion and the structure growth. The Baryon
Oscillation Spectroscopic Survey (BOSS, 2008–2014; Dawson et al., 2012) is the
largest project in the third-stage Sloan Digital Spectroscopic
Survey111http://www.sdss.org/ (SDSS-III; Eisenstein et al., 2011). It has
collected the spectra of more than 1.5 million Luminous Red Galaxies (LRGs;
Reid et al., 2016), the brightest red galaxies in the Universe. Its extended
version, eBOSS (2014–2020; Dawson et al., 2016) in SDSS-IV (Blanton et al.,
2017) has probed another 300,000 LRGs (Ross et al., 2020). Additionally, eBOSS
has observed around 270,000 Emission Line Galaxies (ELGs; Raichoor et al.,
2020). They are bluer, star-forming galaxies that are abundant in the redshift
range of $0.5<z<2$, where there are active star formation processes (Madau et
al., 1998). With such a big amount of tracers, both projects have achieved a
percent-level precision in cosmological parameter measurements (Alam et al.,
2017, 2021a). Ongoing surveys like the Dark Energy Spectroscopic Instrument
(DESI, 2020-2025; DESI Collaboration et al., 2016) is expected to provide
tracers in a larger footprint and with higher-resolution spectra, and reach a
higher precision in scientific results.

Meanwhile, $N$-body simulations can solve numerically the gravitational
evolution equation of DM in $\Lambda$CDM. They are able to accurately describe
the DM distribution in the non-linear regime down to individual haloes in a
large volume and provide halo properties at any cosmic epoch. Thus $N$-body
simulations help validate cosmological theories that are based on perturbation
theories across all the scales, and makes it possible to compare theoretical
predictions with observations.

But there is a gap between dark-matter-only $N$-body simulations and
observations. Since baryonic matter interacts with each other through all
fundamental forces, their clustering properties do not necessarily follow
those of DM on all scales (White & Rees, 1978). Moreover, the spatial
distributions of different types of galaxies, such as LRGs and ELGs, can also
be different in the cosmic web (e.g., Malavasi et al., 2016; Kraljic et al.,
2018). To bridge the gap, a galaxy–halo relation that modulates the clustering
of haloes to match the observation is of great importance. Because it enables
the direct comparison between the theory and the observation via simulation
(Wechsler & Tinker, 2018).

SubHalo Abundance Matching is a simple and intuitive empirical algorithm to
model the galaxy–halo relation. As indicated by the name, SHAM makes use of DM
haloes and their substructures that are dubbed subhaloes. The basic assumption
of SHAM is a monotonic relation (not necessarily linear) between the galaxy
luminosity (or stellar mass) and the halo mass (or the circular velocity;
Kravtsov et al., 2004; Vale & Ostriker, 2004; Nagai & Kravtsov, 2005; Conroy
et al., 2006; Behroozi et al., 2010). The algorithm applies the rank-ordering
for both haloes and galaxies based on mass-related properties as just
mentioned, and assigns galaxies to haloes/subhaloes one by one from the most
massive end, until the SHAM galaxy catalogue reaches a desired number density.

Galaxies from this mock catalogue follow the observed or reconstructed
luminosity function (e.g., Tasitsiomi et al., 2004; Conroy et al., 2006) or
stellar mass function (e.g., Guo et al., 2010; Rodríguez-Torres et al., 2016)
by construction. Their two-point statistics are tuned to be consistent with
the observed one by introducing a Gaussian scatter between the galaxy
luminosity and the halo circular velocity (Tasitsiomi et al., 2004) to include
the scatter of the Tully–Fisher relation (Willick et al., 1997; Steinmetz &
Navarro, 1999). Using the peak maximum circular velocity throughout
accumulation history ($V_{\rm peak}$) instead of the halo mass further
improves the performance of SHAM, since the galaxy accretion is free from the
mass-stripping effect of subhaloes after reaching $V_{\rm peak}$ (Trujillo-
Gomez et al., 2011). With all the improvements, SHAM catalogues of Rodríguez-
Torres et al. (2016) predict the galaxy–halo relation that agrees with the
weak-lensing observation from Shan et al. (2017). Other work on SHAM has also
included the effect of halo assembly bias and galaxy formation (e.g., Chaves-
Montero et al., 2016; Contreras et al., 2021). Due to the single-parameter
feature and its good agreement on data, SHAM has become a useful tool in
cosmological studies.

While most of the SHAM studies choose to fit the projected 2PCF which
marginalizes the effect along the line of sight, 2PCF multipoles can provide
extra information in that direction. Due to the Redshift-Space Distortion
(RSD), 2PCF quadrupole in the redshift space is vulnerable to the bias induced
by the redshift uncertainty. This uncertainty mainly comes from the LRG
redshift determination pipeline that uses broad absorption lines to determine
the redshift (Bolton et al., 2012). For eBOSS LRG, Ross et al. (2020) find the
uncertainty measured by the repeat observations can be fit by a Gaussian
function with a dispersion of 91.8 ${\rm km}\,{\rm s}^{-1}$. This is
negligible for large-scale clustering analysis like Baryon Acoustic
Oscillation (BAO) at over $80\,h^{-1}\,{\rm Mpc}$. Meanwhile, Smith et al.
(2020) demonstrates that the redshift uncertainty of eBOSS quasars influences
the quadrupole on $r<60\,h^{-1}\,{\rm Mpc}$, affecting SHAM that mainly
employs 2PCF on $r<40\,h^{-1}\,{\rm Mpc}$.

Moreover, the standard SHAM algorithm described above works well for bright
galaxies with complete samples, as it matches halo masses and galaxy masses
starting from the most massive ones. When the sample is incomplete due to the
survey requirement (e.g. exclude targets with higher luminosity) or when the
tracer is absent in massive haloes due to the quenching process (e.g.,
Kauffmann et al., 2004; Dekel & Birnboim, 2006), this implementation can be
problematic (Favole et al., 2016a; Rodríguez-Torres et al., 2016).

In this paper, we present a general 3-parameter SHAM algorithm for LRGs on
5-25$\,h^{-1}\,{\rm Mpc}$ that considers the redshift uncertainty effects and
the galaxy completeness. In Section 2, we describe the observational data, the
$N$-body simulation and the galaxy mocks for covariance matrices. The
calculation of 2PCFs and projected 2PCFs, the redshift uncertainty
measurements, and the implementation of SHAM are introduced in Section 3.
Section 4 illustrates the good performance of SHAM in fitting the 2PCF,
reproducing the redshift uncertainty, the galaxy incompleteness evolution and
the projected 2PCF of the observation. Finally, we summarize our findings in
Section 5. In our study, we use a flat $\Lambda$CDM cosmology with
$\Omega_{\rm m}$ = 0.31 and $H_{0}=67.7\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}$, and the line-of-sight direction of the comoving space is along the
$Z$-axis.

The BOSS & eBOSS SHAM study is part of the final release of the eBOSS
measurement. Ross et al. (2020); Lyke et al. (2020) describe in detail
catalogues for the large-scale structure analysis. The mock challenge tasks
are completed by Alam et al. (2021c); Avila et al. (2020a); Rossi et al.
(2021); Smith et al. (2020) for systematics estimations. Galaxy mocks for
covariance matrices used for cosmological analysis are constructed by Lin et
al. (2020); Zhao et al. (2021a). The BAO and RSD
measurements222https://www.sdss.org/science/final-bao-and-rsd-measurements are
based on the clustering of LRGs at $0.6<z<1.0$ (Bautista et al., 2021; Gil-
Marín et al., 2020), ELGs at $0.6<z<1.1$ (Raichoor et al., 2020; Tamone et
al., 2020; de Mattia et al., 2021), quasars (QSOs) at $0.8<z<2.2$ (Hou et al.,
2021; Neveux et al., 2020) and also the Ly$\alpha$ forest at $z>2.1$ (du Mas
des Bourboux et al., 2020). Their cosmological interpretations333see
https://www.sdss.org/science/cosmology-results-from-eboss, and
https://svn.sdss.org/public/data/eboss/DR16cosmo/tags/v1_0_1/ in combination
with the final BOSS results and other probes can be found in Alam et al.
(2021b). eBOSS data also permit the analysis with cosmic voids (Aubert et al.,
2020) and multiple tracers (Wang et al., 2020; Zhao et al., 2021b, 2022).

##  2 Data

###  2.1 SDSS Galaxies

We use the galaxy catalogues from the final data releases of SDSS-III BOSS
(DR12; Alam et al., 2015) and SDSS-IV eBOSS (DR16 Ahumada et al., 2020). BOSS
LRGs are composed of two sub-samples: LOWZ at $0.15<z<0.5$ and CMASS at
$0.4<z<0.7$ (Reid et al., 2016). For eBOSS, its LRG sample has redshift at
$0.6<z<1.0$ (Ross et al., 2020). Galaxy samples in the catalogues are pre-
selected using the photometric information to ensure the observed samples are
clean and abundant at a given redshift range for scientific targets. Their
spectra are observed by fibers (York et al., 2000; Gunn et al., 2006; Smee et
al., 2013) and processed by the spectroscopic pipeline in order to determine
their galaxy type and redshifts (Aihara et al., 2011; Bolton et al., 2012).
The catalogues used for clustering analysis (clustering catalogue hereafter)
are composed of the 3D position of tracers, as well as several weights to
eliminate systematics effects on 2PCF measurements.

####  2.1.1 Target Selection

Target selection using the photometric information includes the signal-to-
noise ratio (SNR) selection, the colour selection, the flux limits cut, and
the star exclusion (Reid et al., 2016; Prakash et al., 2016). In general,
those criteria are meant to have high quality LRG spectra, obtain the designed
redshift range and number density, exclude the low-redshift, blue galaxies and
ensure high-redshift successful rate, and remove stars from LRG samples,
respectively. LOWZ at $0.2<z<0.4$ and CMASS $z<0.6$ are expected to be nearly
volume-limited, i.e., they are complete (Reid et al., 2016). The study of
Leauthaud et al. (2016) confirms the LOWZ completeness but also finds CMASS is
not as complete as Reid et al. (2016) describes.

For eBOSS LRG, there is a special magnitude cut, namely

$i\geq 19.9,$ (1)

which is set to avoid the BOSS CMASS galaxies (Prakash et al., 2016). This
lower limit corresponds roughly to an upper limit of the stellar mass, meaning
that the most massive LRGs may have been excluded from the eBOSS samples. We
use $V_{\rm ceil}$ to account for it and we will explain it in detail in
Section 3.3.

####  2.1.2 Repeat Observations

Repeat observations from SDSS aim at testing the reproducibility of the
spectral measurements and obtaining a higher SNR by coadding multiple spectra
(Dawson et al., 2012). We use them to estimate the redshift uncertainty
statistically in Section 3.2. The redshifts of repeated samples from BOSS are
determined by idlspec2d (Bolton et al., 2012) and those from eBOSS LRGs are
determined by redrock 444https://github.com/desihub/redrock. Less than 5% of
galaxies in the clustering catalogues are observed twice, but those galaxies
are good representatives of the clustering galaxies (Section 4.2).

####  2.1.3 Galaxy Weights

The 2PCF contains the information of cosmological parameters and structure
growth in its amplitude. But the amplitude can be easily modified by various
systematics, e.g., photometric systematics, redshift failures, fibre collision
effects and the varying galaxy number density at different redshift (Reid et
al., 2016; Ross et al., 2020). The corresponding weights to remove their
impacts are: $w_{\rm photosys}$ for all the photometric effects, $w_{\rm noz}$
for the redshift failure rate, $w_{\rm CP}$ for the missing close galaxy pairs
due to the fibre collision and $w_{\rm FKP}$ for minimizing the cosmic
variance.

The fibre collision problem arises from the physical size of fibres (62′′)
that defines the minimum separation between two fibres. Hence galaxies with
small angular distances (i.e., close pairs) cannot be observed in a single
exposure. For the BOSS data, the close-pair weight

$w_{\rm CP}=\frac{N_{\rm CP}+N_{\rm good}}{N_{\rm good}},$ (2)

is applied to the nearest tracer of collided targets to nullify the fibre
collision effects. Here $N_{\rm CP}$ is the number of the half-missing tracer
pairs and $N_{\rm good}$ is the number of tracers with good redshifts (Ross et
al., 2020). This nearest-neighbour approach is not able to fully correct the
fibre collision effect (Guo et al., 2012; Rodríguez-Torres et al., 2016).
Nevertheless, our SHAM fitting results are not biased by this effect (see
Appendix A).

eBOSS uses a different weighting scheme. The pairwise-inverse-probability
(PIP) weighting proposed by Bianchi & Percival (2017) is a better way to
account for galaxy pairs with one missing in a single observation and the
angular up-weighting (ANG) scheme introduced by Percival & Bianchi (2017) can
recover missing galaxy pairs with a distance smaller than the size of a fibre.
Mohammad et al. (2020) apply both PIP and ANG weights on eBOSS samples and
provide unbiased galaxy clustering down to 0.1$\,h^{-1}\,{\rm Mpc}$.

The FKP weight is obtained by (Feldman et al., 1994)

$w_{\rm FKP}=\frac{1}{1+\overline{n}(z)P_{0}},$ (3)

where $\overline{n}(z)$ is the number density at redshift $z$ and $P_{0}$ is
the amplitude of the observed power spectrum at $k\approx{0.15}\,h\,{\rm
Mpc}^{-1}$. For LRGs in BOSS/eBOSS, we take $P_{0}={10000}\,h^{-3}\,{\rm
Mpc}^{3}$ (Reid et al., 2016; Ross et al., 2020).

###  2.2 $N$-body Simulation: UNIT

Future galaxy surveys will span to larger cosmological volumes with a deeper
photometry. The resolution and effective volume of the $N$-body simulation
should also keep up with the improvements. In our study, we use the Universe
$N$-body simulations for the Investigation of Theoretical models from galaxy
surveys555http://www.unitsims.org/ (UNIT; Chuang et al., 2019). This is a
high-resolution and large-effective-volume $N$-body simulation that uses the
fixed-amplitude method to suppress the cosmic variance, thus increasing the
effective volume (Angulo & Pontzen, 2016). Its effective volume is over 10
times larger than that of BOSS/eBOSS LRGs. UNIT has $4096^{3}$ particles in
1$\,h^{-3}\,{\rm Gpc}^{3}$ cubic box, and a mass resolution of 1.2$\times
10^{9}$ $\textup{M}_{\odot}$ $h^{-1}$. Its cosmological parameters are
$\Omega_{m}=0.3089,~{}h\equiv H_{0}/100\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}=0.6774,~{}n_{s}=0.9667,~{}\sigma_{8}=0.8147$. The simulation evolves
from $a(t)=0.01~{}(z=99)$ to $a(t)=1~{}(z=0)$, and produces 128 snapshots at
different redshifts. In our study, we use 12 snapshots from $z=0.2760$ to
$z=0.9011$ as shown in Table 1, and we take one simulation box in each
snapshot given the large effective volume of UNIT (see Section 4.1).

###  2.3 Galaxy Mocks for Covariance Matrices

We use galaxy mocks to calculate covariance matrices in Section 3.4. For BOSS,
we use 1200 DR12 Patchy mocks in each galactic cap. For eBOSS, there are 1000
realizations of EZmock mocks for LRGs in each galactic cap. Both of them are
able to reproduce the observational two-point and three-point statistics down
to non-linear scales.

Kitaura et al. (2014) introduces the Patchy code to generate galaxy mocks
based on the Augmented Lagrangian Perturbation Theory (ALPT). They determine
the galaxy properties by apply their own SHAM to the BigMultiDark
simulation666http://www.multidark.org/ (Klypin et al., 2016) and fit the
observational statistics in both real and redshift space. The clustering
properties of Patchy mocks are then calibrated by those of SHAM catalogues.
The fiducial cosmology is a $\Lambda$CDM cosmology with $\Omega_{m}=0.307115$,
$\Omega_{b}=0.048206$, $h=0.6777$, $\sigma_{8}=0.8225$, $n_{s}=0.9611$
(Kitaura et al., 2016).

The Effective Zel’dovich approximation mock (EZmock) proposed by Chuang et al.
(2015) is another fast methodology to generate mock halo or galaxy catalogues,
with the same cosmology as Patchy mocks. EZmock relies on the Zel’dovich
approximation (Zel’Dovich, 1970) to calculate the displacement of DM particles
at any redshift given their initial positions. Tracers are assigned in the
density field using both the deterministic and the stochastic bias model (Zhao
et al., 2021a). Besides covariance matrix calculations, we also use EZmock
mocks with and without systematics to study the impact of the biased 2PCF
monopole on the projected 2PCF in Section 4.4.

##  3 Methods

###  3.1 Galaxy Clustering

The 2PCF, denoted by $\xi$, measures the excess probability of finding a
galaxy pair compared to a random distribution in a given volume. We use the
Landy–Szalay estimator (LS; Landy & Szalay, 1993) which minimises the
variances of the measurements:

$\xi_{\rm LS}=\frac{\rm DD-2DR+RR}{\rm RR},$ (4)

where galaxy–galaxy pairs (DD), random–galaxy pairs (DR) and random–random
pairs (RR) are normalized by the total number of pairs. $\xi$ and the pair
counts can be calculated as a function of the pair separation $s$ and the
cosine of the angle between s and the line-of-sight ($\beta$), i.e.,
$\mu=\rm\cos(\beta)$. By weighting the 2D $\xi(s,\mu)$ with Legendre
polynomials $P_{\ell}(\mu)$, we obtain the 1D $\xi$ multipoles as

$\xi_{\ell}(s)=\frac{2\ell+1}{2}\int_{-1}^{1}\xi(s,\mu)P_{\ell}(\mu){\rm
d}\mu.$ (5)

In our study we only consider the monopole $\xi_{0}$ and quadrupole $\xi_{2}$
at $s\in$ [5, 25]$\,h^{-1}\,{\rm Mpc}$. This is because scales larger than
25$\,h^{-1}\,{\rm Mpc}$ are prone to imaging systematics (Huterer et al.,
2013) while scales smaller than 5$\,h^{-1}\,{\rm Mpc}$ are sensitive to fibre
collision effects. Even with nearest-neighbour close-pair weights $w_{\rm
CP}$, the biases of the 2PCF quadrupole due to fibre collision effects can be
as large as 3$\sigma$ at $\sim$ 5$\,h^{-1}\,{\rm Mpc}$ (Guo et al., 2012;
Rodríguez-Torres et al., 2016). To understand its influence on our SHAM
results, we test a fitting range of [10, 25]$\,h^{-1}\,{\rm Mpc}$ – which is
only mildly affected by fibre collisions – for the BOSS data. It turns out
that the SHAM constraints in this range are consistent with those on $s\in$
[5, 25]$\,h^{-1}\,{\rm Mpc}$ (see Appendix A). So we use the fitting range
$s\in$ [5, 25]$\,h^{-1}\,{\rm Mpc}$ for all BOSS/eBOSS samples hereafter.

We use linear $s$ bins with 1$\,h^{-1}\,{\rm Mpc}$ interval, and 120 linear
$\mu$ bins in the range of [0,1), for most of the 2PCF measurements in this
work. Only for the eBOSS LRGs in different redshift bins, we have 8
logarithmic bins for $\xi_{0}$ and $\xi_{2}$ respectively.

$\xi$ can also be measured as a function of the distance parallel ($\pi$) and
perpendicular ($r_{p}$) to the line-of-sight. By integrating $\xi(r_{p},\pi)$
with respect to $\pi$, one obtains the projected 2PCF as

$w_{p}(r_{p})=\int_{-\pi_{\rm max}}^{\pi_{\rm max}}\xi(r_{p},\pi){\rm d}\pi.$
(6)

We use 8 logarithmic bins for $r_{p}\in$ [5, 25]$\,h^{-1}\,{\rm Mpc}$, but
$\pi_{\rm max}$ will be changed according to our needs as introduced in
Section 4.4. FCFC777https://github.com/cheng-zhao/FCFC blue(Zhao et al. in
preparation) and Corrfunc Python module (Sinha & Garrison, 2019, 2020) are
employed to calculate $\xi_{\ell}(s)$ and $w_{p}$. $\xi_{\ell}(s)$ of eBOSS
are computed with PIP+ANG-weighted pair counts (Mohammad et al., 2020).

Our SHAM galaxy catalogue is built from an $N$-body simulation in a box, so we
convert the position of SHAM galaxies from real space to redshift space before
calculating the 2PCF with (Kaiser, 1987)

$Z_{\rm redshift}=Z_{\rm real}+\frac{v_{\rm pec,Z}}{a(t)H(z)},$ (7)

where $Z$ is the coordinate and $z$ is the redshift, $v_{\rm pec,Z}$ is the
peculiar velocity along the $Z$-axis. The estimator used to calculate the SHAM
2PCF is the Peebles–Hauser estimator (Peebles & Hauser, 1974)

$\xi_{\rm PH}=\frac{\rm DD}{\rm RR}-1,$ (8)

where galaxy–galaxy pairs (DD) is also normalized by the total number of
pairs. The random–random (RR) counts follow the analytical expression:

$\rm
RR=\frac{4\uppi}{3}\frac{\textit{s}_{\text{max}}^{3}-\textit{s}_{\text{min}}^{3}}{\textit{V}_{\text{box}}}\frac{1}{\textit{N}_{\mu}},$
(9)

where $s_{\text{max}}$ and $s_{\text{min}}$ are the boundaries of the
separation bins, $V_{\text{box}}=1\,h^{-3}\,{\rm Gpc}^{3}$ is the volume of
the simulation box and $N_{\mu}=120$ is the number of $\mu$ bins. We do not
apply weights to SHAM galaxies as there are no observational systematics and
radial selection. The $\xi_{\ell}(s)$ of SHAM galaxies is calculated by the
built-in 2PCF calculator in our SHAM implementation code.

###  3.2 Redshift Uncertainty

The redshift uncertainty is an inevitable error while measuring the redshift.
It mainly affects the quadrupole of our clustering measurements (Smith et al.,
2020), since it can be regarded as random peculiar motions along the line-of-
sight. As suggested in Bolton et al. (2012), the redshift determination
pipeline is the primary source of the redshift uncertainty of SDSS LRGs, but
it is not the dominant source. Hence, using the error given by the pipeline
will underestimate the redshift uncertainty. As a result, we estimate it
statistically via repeat observations.

The redshift difference between two measurements $\Delta z$ can be converted
to $\Delta v$ that represents the radial motion as

$\Delta v=\frac{\Delta z}{(1+z)c},$ (10)

where $z$ is the redshift of one of measurements with
$\textsc{SPECPRIMARY}=1$. With jackknife errors, we fit all the $\Delta v$
histograms with Gaussian distributions, obtaining the best-fitting dispersion
$\sigma_{\Delta v}$ as illustrated in Fig. 1. We also quote their standard
deviation, $\hat{\sigma}_{\Delta v}$, as a statistical estimate of the
redshift uncertainty. In our SHAM algorithm, its clustering effect is
quantified by $v_{\rm smear}$.

Figure 1: The Gaussian fitting of the $\Delta v$ histograms. The error bars of
$\Delta v$ histograms are calculated from jackknife re-sampling. The best-
fitting Gaussian models are shown in red lines, labelled with their dispersion
values $\sigma_{\Delta v}$. The standard deviations of $\Delta v$ is written
below the legend and the proportion of outliers that are beyond the Gaussian
fitting range is indicated above the subplots. SDSS has a Gaussian $\Delta v$
histogram, and the ourlier rate of eBOSS are much larger than those of BOSS,
consistent with the difference between $\sigma_{\Delta v}$ and
$\hat{\sigma}_{\Delta v}$.

###  3.3 SHAM Implementation

Recent SHAM studies (e.g., Campbell et al., 2018; Granett et al., 2019;
Contreras et al., 2021) consistently choose $V_{\rm peak}$ to represent the
halo mass of haloes and subhaloes. The subsequent the progress of the SHAM
algorithm is to sort the galaxy stellar mass and $V_{\rm peak}^{\rm scat}$,
which is obtained as

$V_{\rm peak}^{\rm scat}=V_{\rm peak}S_{g},$ (11)

where

$S_{g}=1+\mathcal{N}(0,\sigma^{2}),$ (12)

and match them one by one from the largest $V_{\rm peak}^{\rm scat}$, until
the number density of SHAM galaxies equals to that of the observation. In Eq.
(11), $\mathcal{N}(0,\sigma^{2})$ represents a Gaussian random distribution
centred on zero with $\sigma^{2}$ variance, where $\sigma$ is the only free
parameter in the SHAM. The choice of the Gaussian function is due to the fact
that the Tully–Fisher relation, i.e., the galaxy–halo mass relation, has
Gaussian distributed residuals (Willick et al., 1997; Steinmetz & Navarro,
1999). We assert

$S_{g}=\exp(\mathcal{N}(0,\sigma^{2})).$ (13)

when the Gaussian scatter is negative. Nevertheless, it does not change the
fitting results.

There are four prerequisites for the standard SHAM described above: 1) the
cosmology of the simulation has to be close to the true one; 2) the simulation
should have a good resolution to resolve the subhaloes down to the low-mass
end and halo properties should be accurate; 3) the observed clustering
measurements should be unbiased; 4) the observational stellar mass function
(SMF) has to be complete in the massive end. If one of these prerequisites is
missing, SHAM will not model the observed clustering signal accurately on all
scales.

For eBOSS LRGs, there is a rough truncation of the stellar mass in the massive
end (Section 2.1.1). A straightforward way to account for this massive-end
incompleteness is to discard the most massive haloes. We introduce a new
parameter $V_{\rm ceil}$ to remove the $V_{\rm ceil}$ per cent of the largest
scattered $V_{\rm peak}$ (i.e., $V^{\rm scat}_{\rm peak}$). Since $\sigma$ and
$V_{\rm ceil}$ both modulate the amplitude of 2PCF monopole, they are
degenerate.

The redshift uncertainty can add bias to the observed 2PCF quadrupole.
Therefore, we introduce an extra Gaussian-random distribution
$\mathcal{N}(0,v_{\rm smear}^{2})$ in the peculiar velocity of SHAM galaxies
as

$v^{\rm scat}_{\rm pec,Z}=v_{\rm pec,Z}+\mathcal{N}(0,v_{\rm smear}^{2}).$
(14)

The target is to mimic the impact of the measured Gaussian redshift
uncertainties in the redshift-space clustering as presented in Fig. 1.

To conclude, our 3-parameter SHAM implementation is as follows:

  1. 1.

Scatter the $V_{\rm peak}$ as suggested in Eq. (11) with

$S_{g}=\left\\{\begin{array}[]{ll}1+\mathcal{N}(0,\sigma^{2}),&\mathcal{N}(0,\sigma^{2})\geq
0\\\ \exp(\mathcal{N}(0,\sigma^{2})),&\mathcal{N}(0,\sigma^{2})<0;\\\
\end{array}\right.$ (15)

  2. 2.

Sort $V^{\rm scat}_{\rm peak}$ and discard the most massive $V_{\rm ceil}$ per
cent of haloes/subhaloes;

  3. 3.

From the remaining catalogue, keep the $N_{\rm gal}$-th largest $V^{\rm
scat}_{\rm peak}$;

  4. 4.

Assign galaxies in the centre of those haloes/subhaloes, and smear the
peculiar velocities of galaxies along the line of sight with Eq. (14);

  5. 5.

Convert the galaxy coordinates from the real space to the redshift space using
Eq. (7).

To model correctly the observed clustering, the redshift of the UNIT snapshot
should be close to the effective redshift of the data computed with

$z_{\rm eff}=\frac{\sum zw^{2}}{\sum w^{2}}.$ (16)

where $w$ is the total galaxy weight. For eBOSS (Ross et al., 2020),

$w=w_{\rm photosys}w_{\rm FKP}w_{\rm noz}w_{\rm CP},$ (17)

while for BOSS the total weight has a different form (Reid et al., 2016)

$w=w_{\rm photosys}w_{\rm FKP}(w_{\rm noz}+w_{\rm CP}-1).$ (18)

Then we apply SHAM on this snapshot to find the best-fitting parameters.
Besides the 2PCF calculator mentioned in Section 3.1, our code also has a
built-in MultiNest sampler which will be introduced in Section 3.4.

The number of SHAM galaxies $N_{\rm gal}$ is determined by the observed
effective number density $n_{\rm eff}$ in a given redshift range calculated as

$\begin{split}n_{\rm eff}&=\sqrt{\frac{\int n(z)^{2}{\rm d}V}{\int{\rm
d}V}}=\sqrt{\frac{\int n(\chi)^{2}\chi^{2}{\rm d}\chi}{\int\chi^{2}{\rm
d}\chi}},\end{split}$ (19)

where the second equality is due to ${\rm d}V=A_{\text{eff}}((\chi+{\rm
d}\chi)^{3}-\chi^{3})/3=A_{\text{eff}}\chi^{2}{\rm d}\chi+\mathcal{O}({\rm
d}\chi^{2})$, $A_{\text{eff}}$ is the effective area of the footprint and
$\chi$ is the comoving distance for an object at redshift $z$. The values are
reported in Table 1.

Figure 2: The best-fitting 3-parameter SHAM 2PCF multipoles compared with those of the LOWZ observations on 5–25$\,h^{-1}\,{\rm Mpc}$. The observational data and the cosmic variance $\epsilon_{\rm obs,\xi_{0,2}}$ are represented by dots with error bars. Solid lines with shades are SHAM 2PCFs and their errors. The first and third row are the monopole and the quadrupole respectively. Residuals normalized by $\epsilon_{\rm obs,\xi_{0,2}}$ are shown in the second and fourth row. The left column is for galaxy samples in redshift slices and the right column is for those in the bulk redshift range. SHAM 2PCFs agree with the observations with rescaled reduced $\chi^{2}$ values around 1. Figure 3: Same figure as Fig. 2, but for CMASS LRGs. The deviation for the bulk CMASS sample (right panels) is relatively large compared to those of the sliced redshift bins. We attribute this difference to its inhomogeneous sample completeness shown in Section 4.1. Figure 4: Same figure as Fig. 2, but for eBOSS LRGs. The SHAM 2PCFs also agree with the observations except the one at $0.7<z<0.9$. project | redshift | $z_{\rm eff}$ | $z_{\rm UNIT}$ | $V_{\rm eff}$ | 10${}^{4}n_{\rm eff}$ | $\sigma$ | $v_{\rm smear}$ | $V_{\rm ceil}$ |  $\chi^{2}$/dof | rescaled
---|---|---|---|---|---|---|---|---|---|---
| range |  |  | $(h^{-3}\,{\rm Gpc}^{3})$ | $(h^{3}\,{\rm Mpc}^{-3})$ |  | $({\rm s}^{-1}\,{\rm km})$ | $(\%)$ |  |  $\chi^{2}$/dof
LOWZ | $0.2<z<0.33$ | 0.2754 | 0.2760 | 0.29 | 3.37 | $0.09^{+0.06}_{-0.06}$ | $100.3^{+7.6}_{-10.4}$ | $0.0067^{+0.0019}_{-0.0025}$ | 32/37 | 31/37
$0.27^{+0.02}_{-0.02}$ | $66.1^{+8.2}_{-9.8}$ | / | 33/38 | 32/38
LOWZ | $0.33<z<0.43$ | 0.3865 | 0.3941 | 0.33 | 2.58 | $0.20^{+0.05}_{-0.05}$ | $73.9^{+12.2}_{-12.3}$ | $0.0059^{+0.0026}_{-0.0028}$ | 51/37 | 50/37
$0.34^{+0.02}_{-0.01}$ | $30.5^{+16.7}_{-16.2}$ | / | 54/38 | 52/38
CMASS | $0.43<z<0.51$ | 0.4686 | 0.4573 | 0.47 | 3.42 | $0.30^{+0.05}_{-0.08}$ | $45.0^{+20.6}_{-16.6}$ | $0.0078^{+0.0074}_{-0.0040}$ | 41/37 | 39/37
$0.43^{+0.01}_{-0.02}$ | $17.4^{+11.8}_{-9.6}$ | / | 45/38 | 43/38
CMASS | $0.51<z<0.57$ | 0.5417 | 0.5574 | 0.46 | 3.63 | $0.23^{+0.04}_{-0.04}$ | $23.2^{+18.5}_{-15.0}$ | $0.0144^{+0.0037}_{-0.0036}$ | 43/37 | 41/37
$0.42^{+0.02}_{-0.01}$ | $6.4^{+6.2}_{-4.0}$ | / | 60/38 | 58/38
CMASS | $0.57<z<0.7$ | 0.6399 | 0.6281 | 0.65 | 1.60 | $0.17^{+0.13}_{-0.02}$ | $73.1^{+8.2}_{-25.3}$ | $0.0459^{+0.0029}_{-0.0143}$ | 54/37 | 51/37
eBOSS | $0.6<z<0.7$ | 0.6518 | 0.6644 | 0.16 | 0.939 | $0.58^{+0.29}_{-0.22}$ | $100.4^{+12.3}_{-12.8}$ | $0.0510^{+0.0324}_{-0.0194}$ | 16/13 | 16/13
eBOSS | $0.6<z<0.8$ | 0.7071 | 0.7018 | 0.33 | 0.886 | $0.38^{+0.31}_{-0.13}$ | $98.8^{+10.8}_{-13.4}$ | $0.0617^{+0.0185}_{-0.0308}$ | 24/13 | 23/13
eBOSS | $0.7<z<0.9$ | 0.7968 | 0.8188 | 0.26 | 0.647 | $0.09^{+0.17}_{-0.04}$ | $128.5^{+9.5}_{-22.2}$ | $0.0690^{+0.0044}_{-0.0172}$ | 30/13 | 30/13
eBOSS | $0.8<z<1.0$ | 0.8778 | 0.9011 | 0.09 | 0.301 | $0.22^{+0.17}_{-0.14}$ | $134.2^{+18.5}_{-20.1}$ | $0.0481^{+0.0153}_{-0.0155}$ | 16/13 | 16/13
LOWZ | $0.2<z<0.43$ | 0.3441 | 0.3337 | 0.62 | 2.95 | $0.22^{+0.03}_{-0.05}$ | $76.6^{+10.2}_{-8.2}$ | $0.0031^{+0.0023}_{-0.0015}$ | 45/37 | 42/37
$0.30^{+0.01}_{-0.01}$ | $54.5^{+6.9}_{-6.7}$ | / | 45/38 | 42/38
CMASS | $0.43<z<0.7$ | 0.5897 | 0.5924 | 1.58 | 2.64 | $0.20^{+0.20}_{-0.03}$ | $63.5^{+9.1}_{-44.6}$ | $0.0269^{+0.0032}_{-0.0190}$ | 81/37 | 70/37
eBOSS | $0.6<z<1.0$ | 0.7781 | 0.7018 | 0.43 | 0.626 | $0.40^{+0.19}_{-0.06}$ | $109.7^{+8.0}_{-6.1}$ | $0.0542^{+0.0067}_{-0.0196}$ | 35/37 | 33/37
Table 1: Properties of analysed galaxy samples and their best-fitting SHAM
results. The first six columns are the project names of galaxy samples, their
redshift ranges, effective redshifts $z_{\rm eff}$, the closest redshifts of
the UNIT snapshot to $z_{\rm eff}$, effective volumes, and their average
number densities. The next four columns are the constraints of SHAM parameters
{$\sigma,v_{\rm smear},V_{\rm ceil}$} and the minimum $\chi^{2}$/dof. The
final column is the rescaled $\chi^{2}$/dof taking into account the
uncertainty of UNIT simulations. There are redshift bins with two sets of SHAM
parameters. The first set (row) is from the 3-parameter SHAM and the second
set is from 2-parameter SHAM without $V_{\rm ceil}$. The 2-parameter SHAM can
only be applied to complete samples, i.e., LOWZ and CMASS galaxies at $z<0.6$
(Leauthaud et al., 2016). Redshift bins with only one set of SHAM results are
from the 3-parameter SHAM.

###  3.4 SHAM Fitting

The SHAM best-fitting parameters are obtained by minimizing the $\chi^{2}$
value (i.e., the maximum log-likelihood) between the SHAM and the
observational 2PCF. The $\chi^{2}$ for a given parameter set
$\Theta=\\{\sigma,v_{\rm smear},V_{\rm ceil}\\}$ is defined as

$\chi^{2}(\Theta)=(\xi_{\rm data}-\xi_{\rm model}(\Theta))^{T}{\bf\it
C}^{-1}(\xi_{\rm data}-\xi_{\rm model}(\Theta)),$ (20)

where $\xi=(\xi_{0},\xi_{2})$ denotes the vector composed of the 2PCF monopole
and quadrupole. The vectors $\xi_{\rm data}$ and $\xi_{\rm model}$ represent
the data vector and the SHAM model 2PCF respectively. In particular, $\xi_{\rm
model}$ is obtained by averaging the 2PCFs of 32 SHAM galaxy realizations
generated using the same $\Theta$ with different random seeds. This operation
reduces the SHAM statistical uncertainty. ${\bf\it C}$ is the unbiased
covariance matrix (Hartlap et al., 2007):

${\bf\it C}^{-1}={\bf\it C}^{-1}_{s}\frac{N_{\rm m}-N_{s}-2}{N_{\rm m}-1},$
(21)

where $N_{s}$ is the length of $\xi_{\rm data}$, i.e., the total number of
bins used in the 2PCF fitting, and $N_{\rm m}$ is the number of mocks used to
compute the covariance matrix. $N_{\rm m}=1200$ for BOSS Patchy mocks and
$N_{\rm m}=1000$ for eBOSS EZmock mocks. ${\bf\it C}_{s}$ is the covariance
matrix calculated as (Zhao et al., 2021a)

${\bf\it C}_{s,\rm ij}=\frac{1}{\rm N_{\rm m}-1}\sum^{\rm N_{\rm
m}}_{k=1}[\xi_{\rm k}(s_{\rm i})-\overline{\xi}(s_{\rm i})][\xi_{\rm k}(s_{\rm
j})-\overline{\xi}(s_{\rm j})],$ (22)

where $\xi_{\rm k}$ is the correlation function measured from the $k_{\rm th}$
mock, and $\overline{\xi}(s_{\rm i})$ is the average of the mock correlation
function in a given distance bin $s_{\rm i}$.

We assume a Gaussian likelihood $\mathcal{L}(\Theta)$ for our parameters,
which is

$\mathcal{L}(\Theta)\propto\rm e^{-\frac{\chi^{2}(\Theta)}{2}},$ (23)

and employ a Monte-Carlo sampler
Multinest888https://github.com/farhanferoz/MultiNest (Feroz & Hobson, 2008;
Feroz et al., 2009; Feroz et al., 2019), an efficient nested sampling
technique especially for multi-modal posteriors, to constrain $\Theta$. The
survival volume (i.e., the prior volume) is defined as (Feroz & Hobson, 2008)

$X(\lambda)=\int_{\\{\Theta:\mathcal{L}(\Theta)>\lambda\\}}\text{Pr}(\Theta|H){\rm
d}\Theta,$ (24)

where $\text{Pr}(\Theta|H)$ is the parameter prior, and the evidence integral
can be written as

$\mathcal{Z}=\int_{0}^{1}\mathcal{L}(\Theta){\rm
d}X\approx\sum^{M}_{i=1}\mathcal{L}_{i}w_{i},$ (25)

where $i$ is the number of iterations, $w_{i}=\frac{1}{2}(X_{i-1}-X_{i+1})$,
and $X_{i}$ is a sequence of decreasing values as
$0<X_{M}<...<X_{2}<X_{1}<X_{0}=1$. During the sampling, live points walk
randomly and simultaneously in the parameter space confined by the prior, and
some of them will be deactivated if they have the lowest likelihood
$\mathcal{L}_{i}$. The sampling terminates if the evidence contribution from
the $j_{\rm th}$ iteration $\Delta\mathcal{Z}_{j}=\mathcal{L}_{\rm max}X_{i}$
is lower than a certain threshold (i.e., the tolerance), where
$\mathcal{L}_{\rm max}$ is the maximum likelihood among the current set of
live points (Feroz et al., 2019). The final results of our SHAM do not change
much when changing the tolerance and the initial number of live points.
Therefore, for all the tests, we set the tolerance to be 0.5 and the number of
particles to be 200, in order to improve the efficiency of the convergence.

The built-in analyser of
pyMultinest999https://github.com/JohannesBuchner/PyMultiNest (Buchner et al.,
2014) is used to extract the median value and the 1$\sigma$ limits of the
parameters, and the maximum-likelihood $\chi^{2}$ listed in Table 1. The
parameter set with the maximum-likelihood $\chi^{2}$ is indicated in the
posteriors provided by Getdist (Lewis, 2019) in Appendix C.

##  4 Results

Figure 5: 2PCF multipoles of the best-fitting 2-parameter SHAM (with $\sigma$
and $v_{\rm smear}$) compared with those of the observations on
5–25$\,h^{-1}\,{\rm Mpc}$ as in Fig. 2. The first column is for LOWZ in
redshift slices, the second column is for LOWZ at a bulk redshift range, and
the third column is for CMASS in redshift slices. SHAM 2PCF at $z<0.51$ agrees
with the observational 2PCF. However, a significant difference in the first
two bins of the quadrupole is shown for the CMASS sample at $0.51<z<0.57$.

###  4.1 Galaxy Clustering Fitting

We calibrate our SHAM model with LRGs on 5–25$\,h^{-1}\,{\rm Mpc}$ for the
full LOWZ, CMASS and eBOSS LRG samples as well as 9 finer redshift bins as
summarized in Table 1. Figures 2–4 show that 2PCFs of the SHAM catalogues
agree with those of the observational data in general. Note that there are 32
realizations of SHAM catalogues produced by the maximum-likelihood parameter
set, aiming at reducing the statistical error brought by the Gaussian
scatters. For this reason, we choose the mean value of their 2PCFs to be the
SHAM 2PCF, and rescale their 2PCF standard deviations by $1/\sqrt{32}$ to
obtain the error of the SHAM. We find that the SHAM error is negligible
compared to observed statistical error determined by PATCHY mocks or EZmock
mocks. Thus, we decided to ignore it in this study.

The reduced $\chi^{2}$ values from the fits do not include cosmic variances of
the UNIT simulations. To include their influence in the analysis, we divide
the reduced $\chi^{2}$ by

$\frac{\epsilon^{2}_{\rm com}}{\epsilon^{2}_{\rm obs}}=1+\frac{\epsilon_{\rm
UNIT}^{2}}{\epsilon_{\rm obs}^{2}}\approx 1+\frac{V_{\rm eff,obs}}{V_{\rm
eff,UNIT}},$ (26)

where $\epsilon^{2}_{\rm obs}$ and $\epsilon^{2}_{\rm UNIT}$ represent the
observational variance and the cosmic variance of UNIT simulation respectively
and $\epsilon^{2}_{\rm com}$ is the combination of two errors. $V_{\rm
eff,obs}$ is the effective volume calculated with (Wang et al., 2013; Reid et
al., 2016)

$V_{\rm
eff,obs}=\sum_{i}\left(\frac{\bar{n}(z_{i})P_{0}}{1+\bar{n}(z_{i})P_{0}}\right)^{2}\Delta
V(z_{i}),$ (27)

where $\Delta V(z_{i})$ is the volume of the shell at redshift $z_{i}$ and
$\overline{n}(z_{i})$ is the mean number density of that shell. The values are
listed in the fourth column of Table 1 and we assume $V_{\rm
eff,UNIT}=10\,h^{-3}\,{\rm Gpc}^{3}$ (Chuang et al., 2019). The rescaled
$\chi^{2}$ values are recorded in the final column of Table 1. As the rescaled
value is a better indicator of the goodness of the fit than the $\chi^{2}$
value, our discussions hereafter are all based on the rescaled ones.

The $\chi^{2}$ value for the bulk CMASS sample is relatively large compared to
those of CMASS redshift slices and other bulk samples. This can be attributed
to the varying sample completeness of CMASS sample which we will prove later
in this chapter. The 2PCF quadrupoles of all CMASS SHAM samples are
underestimated on 18–25$\,h^{-1}\,{\rm Mpc}$, which is consistent with the
results of Rodríguez-Torres et al. (2016). Similar (but less obvious)
discrepancies are also observed from the eBOSS samples. They may be due to
some uncorrected systematics. The monopole disagreement on $r>10\,h^{-1}\,{\rm
Mpc}$ and the resulting large $\chi^{2}$ value for eBOSS SHAM at $0.7<z<0.9$
might be due to observational systematics as well.

Our measured $\sigma\in[0.17,0.40]$ for CMASS at $0.43<z<0.7$ is consistent
with $\sigma=0.31$ obtained by Rodríguez-Torres et al. (2016). It is worth
noting that their scatter considers the incompleteness of the observed stellar
mass function, i.e., they obtain the intrinsic scatter $\sigma_{\rm int}$. But
we assume that the observed stellar mass function in a certain range is
complete. It means that our $\sigma$ account for both the intrinsic scatter
and the observed incompleteness.

Since BOSS samples are complete at $z<0.6$ (Reid et al., 2016; Leauthaud et
al., 2016), we also apply our SHAM model without $V_{\rm ceil}$ (2-parameter
SHAM hereafter) to the same simulation snapshot as the 3-parameter version and
fit to the same data. The best-fitting 2PCFs and parameter constraints are
shown in Fig. 5 and Table 1 respectively. The rescaled reduced $\chi^{2}$
values for the 2- and 3-parameter SHAM models are similar for all LOWZ LRGs
and CMASS LRGs at $0.43<z<0.51$, which confirms their galaxy completeness. The
reduced $\chi^{2}$ for the 2-parameter SHAM at $0.51<z<0.57$ is significantly
larger (the $\chi^{2}$ difference shows that SHAM with $V_{\rm ceil}=0$ is
rejected more than 4$\sigma$), due to the discrepancy of quadrupole on
5–7$\,h^{-1}\,{\rm Mpc}$. It also means $\sigma$ and $V_{\rm ceil}$ are not
completely degenerate. The monopole of the best-fitting 2-parameter SHAM is
also systematically lower than that of the observation. The issue in the
result of the 2-parameter SHAM demonstrate that CMASS at $0.51<z<0.57$ is not
complete, i.e., $V_{\rm ceil}\neq 0$, contrary to what has been presented in
Reid et al. (2016). This is consistent with the marginalized posterior
distributions of $V_{\rm ceil}$ in the 3-parameter SHAM (see Fig. 20).

###  4.2 Redshift Uncertainty indicated by $v_{\rm smear}$

Figure 6: Comparison between the measured $v_{\rm smear}$ from the 3-parameter
SHAM (black dots with error bars) and that from the 2-parameter SHAM (open
circles with error bars; a horizontal offset is applied to avoid overlapping),
the best-fitting Gaussian dispersion $\sigma_{\Delta v}$ (blue line with
shades) and the standard deviations $\hat{\sigma}_{\Delta v}$ (red lines with
shades) of the $\Delta v$ distribution. The left panel is for the entire
sample, while the right panel is for samples in redshift slices. Both the BOSS
CMASS sample and the eBOSS LRG sample show good agreements between $v_{\rm
smear}$ values and the redshift uncertainties measured from repeat samples.
However, for the BOSS LOWZ sample, $v_{\rm smear}$ values are significant
larger than those estimated from repeat observations. Figure 7: Comparison
among the quadrupoles from the observation (red dots with error bars), the
best-fitting SHAM (red lines), and the SHAM 2PCF obtained by replacing the
best-fitting $v_{\rm smear}$ with $\sigma_{\Delta v}$ (black lines). The
first, second and third column represents LOWZ samples at $0.2<z<0.33$,
$0.33<z<0.43$ and $0.2<z<0.43$, respectively. The differences between two
quadrupoles on $r<10\,h^{-1}\,{\rm Mpc}$ are mostly larger than 3$\sigma$.
Figure 8: Same as Fig. 7, but using the LOWZ NGC sample (magenta lines) and
LOWZ SGC sample (blue lines) separately. The discrepancies between $v_{\rm
smear}$ and $\sigma_{\Delta v}$ are not mitigated for LOWZ SGC when there is a
high rate of SDSS-III sample.

Fig. 6 shows that $\hat{\sigma}_{\Delta v}$ increases monotonically with
$z_{\rm eff}$. This is consistent with the fact that the absorption lines used
to determine the redshift are broader at higher redshift than those at lower
redshift, leading to larger redshift uncertainties. There is a general
consistency between the best-fitting SHAM $v_{\rm smear}$ and
$\hat{\sigma}_{\Delta v}$, $\sigma_{\Delta v}$ for CMASS. For the eBOSS
samples, SHAM $v_{\rm smear}$ agrees with $\hat{\sigma}_{\Delta v}$, but both
of them are systematically larger than $\sigma_{\Delta v}$. It means that we
cannot neglect the Gaussian-fitting outliers and their effects on eBOSS
quadrupoles. The tail of eBOSS $\Delta v$ distributions also slightly deviates
from the Gaussian model, meaning that eBOSS $\sigma_{\Delta v}$ is not a good
representative of the observed redshift uncertainty. The best-fitting $v_{\rm
smear}$ values of 2-parameter SHAM are lower than those of the 3-parameter
SHAM, and this difference is larger for complete samples at $z<0.51$ than that
of CMASS at $0.51<z<0.57$. This is not a contradiction because the biggest
change in $\sigma$–$v_{\rm smear}$ posterior happens in CMASS at $0.51<z<0.57$
as shown in Figures 23–25. The small change in $v_{\rm smear}$ is due to its
weaker $V_{\rm ceil}$–$v_{\rm smear}$ degeneracy.

Discrepancies also exist for LOWZ LRGs at all redshifts and they remain for
the 2-parameter LOWZ SHAM. To compare the differences in the quadrupole, we
replace the best-fitting $v_{\rm smear}$ values with $\sigma_{\Delta v}$,
generate SHAM galaxy catalogues and calculate the 2PCFs. As presented in Fig.
7, their quadrupole differences are larger than 3$\sigma$. It means that
besides the statistical redshift uncertainty, LOWZ has unknown factors that
boost the quadrupole. We investigate various potential causes listed below.

A. Missing subhaloes or velocity bias in the simulation: The host
haloes/subhaloes for LRGs have a large mass (e.g. $>1000$ particles), so we
expect they are well resolved. But their peculiar velocity might have biases
especially for subhaloes. However, our simulation-based SHAM provides $v_{\rm
smear}$ that agrees with CMASS and eBOSS samples and we do not expect a
significant evolution in the velocity bias. Therefore, the properties of
haloes and subhaloes should be reliable and do not lead to the $v_{\rm smear}$
disagreement in LOWZ.

B. Observational systematics: The LOWZ sample includes SDSS-I/II galaxies with
a redshift determination pipeline different from that of SDSS-III BOSS (Bolton
et al., 2012). Thus, it is possible that the spectroscopic pipelines yield
different uncertainties of redshift measurements. In fact, we find that in the
clustering catalogue, 56 per cent of galaxies in the North Galactic Cap (NGC)
and 90 per cent of galaxies in the South Galactic Cap (SGC) are from SDSS-III.
The component difference in two galactic caps indicates that we can analyse
NGC and SGC separately. Repeated samples are all from SDSS-III. Therefore, the
$\Delta v$ from repetitive observations may not be representative of the
redshift uncertainty of the LOWZ clustering.

However, as shown in Fig. 8, a high proportion of SDSS-III galaxies in the SGC
does not mean a smaller difference between the $\sigma_{\Delta v}$-generated
2PCF and the $v_{\rm smear}$-generated 2PCF compared to the difference for
LOWZ in the NGC. Therefore, the difference in the two spectroscopic pipelines
cannot explain the $v_{\rm smear}$ discrepancy.

C. Underestimation of redshift uncertainties: Considering the fact that errors
from the spectroscopic pipeline underestimate redshift uncertainties, it is
difficult to directly prove the representative of samples from the repeat
observation. Nevertheless, we can proceed with the photometric information
(Appendix B). In the LOWZ sample, we find that the fraction of the blue
samples $f_{\rm blue}$ in repeat observation is smaller than that of the
clustering sample, but the differences are minor (See Table 2 and Table 3).
Given the similarity in the $f_{\rm blue}$ and colour–redshift distributions,
the sample of the repeat observations are unbiased representatives of galaxies
in the clustering catalogue. Even if the repeated samples are biased, this is
not the main cause of the discrepancy. Because the redshift uncertainty of
LOWZ at lower redshift should be smaller than that of CMASS, but $v_{\rm
smear}$ of LOWZ is systematically larger than that of CMASS.

D. A more complete model for galaxy assignment: While we are using the same
model for CMASS and eBOSS LRGs to describe LOWZ, the discrepancy might
indicate that a more complete model is required. E.g., the LOWZ sample is
composed of a different type of galaxies which has more satellites than CMASS.

In fact, our satellite fraction ($f_{\rm sat}$) for LOWZ at $0.2<z<0.43$ is
12.6 per cent, similar to CMASS $f_{\rm sat}=12.3$ per cent at $0.43<z<0.7$.
They are close to results from Halo Occupation Distribution fitting with
$w_{p}$, another empirical model for the galaxy–halo relation: Parejko et al.
(2013) based on DR9 give $12\pm 2$ per cent for LOWZ at $0.2<z<0.4$ and Reid
et al. (2014) based on DR10 give $10\pm 2$ per cent at $0.43<z<0.7$. Both SHAM
and HOD do not support the explanation of large $f_{\rm sat}$ in LOWZ.
Nevertheless, given the difference in data and fitting scales, $f_{\rm sat}$
can still be the explanation. Additionally, as pointed out by Ross et al.
(2014) and Favole et al. (2016b), we know that the blue tail of CMASS galaxies
show a higher quadrupole at 5–40$\,h^{-1}\,{\rm Mpc}$ compared to the
dominating redder galaxies, which has 2PCF close to that of the full
catalogue. For LOWZ, the ad hoc blue galaxies might play a more important role
in the clustering properties. The velocity bias is also found to be different
in CMASS and LOWZ samples (Guo et al., 2015; Lange et al., 2022). So it is
possible that the LOWZ discrepancy can be mitigated by a better model that
considers those factors. We leave it for a future work.

Figure 9: The redshift evolution of the completeness of eBOSS LRG targets. The
photometric redshifts ($z_{\rm photo}$) are taken from the DECaLS DR9
catalogues. The increasing completeness with respect to the redshift is the
result of the i-band lower limit imposed in the LRG target selection.  Figure
10: The $w_{p}$ comparison between the best-fitting SHAM catalogues and the
observations. The first row and the third row are $w_{p}$ with $\pi$
integrating up to 80$\,h^{-1}\,{\rm Mpc}$ and 30$\,h^{-1}\,{\rm Mpc}$
respectively, and their residuals normalized by the cosmic variance
$\epsilon_{\rm obs,w_{p}}$ is shown in the second and fourth row. $\pi_{\rm
max}$ in the figures is for both observational and SHAM galaxies. The
deviation at $r_{p}$ around 20$\,h^{-1}\,{\rm Mpc}$ for the bulk sample
($0.2<z<0.43$) decreases while using a smaller $\pi_{\rm max}$. However, we
see that the deviation at 5$\,h^{-1}\,{\rm Mpc}$ increases which might
indicate systematics at smaller scales. Figure 11: Same as Fig. 10, but for
CMASS LRGs. SHAM catalogues start to be more consistent with the observed
$w_{p}$ with $\pi_{\rm max}=$ 30$\,h^{-1}\,{\rm Mpc}$ compared to $\pi_{\rm
max}=$ 80$\,h^{-1}\,{\rm Mpc}$. Large deviations are found in the case of
$\pi_{\rm max}=$ 80$\,h^{-1}\,{\rm Mpc}$. This is consistent with our findings
from multipoles which indicate systematics at larger scales, e.g.
$r>18\,h^{-1}\,{\rm Mpc}$. Figure 12: Same as Fig. 10, but for eBOSS LRGs.
Left figures (small redshift bins) show obvious deviations when using
$\pi_{\rm max}=$ 80$\,h^{-1}\,{\rm Mpc}$, but the deviations are mitigated
when using $\pi_{\rm max}=$ 30$\,h^{-1}\,{\rm Mpc}$.

###  4.3 Galaxy Incompleteness indicated by $\sigma$ and $V_{\rm ceil}$

Due to the high degeneracy between $\sigma$ and $v_{\rm ceil}$, the redshift
evolutions of $\sigma$ and $V_{\rm ceil}$ are not obvious in the marginalized
posterior of eBOSS (Fig. 22). But the 2D $\sigma$–$V_{\rm ceil}$ posterior
contours show a clear trend that $\sigma$ or $V_{\rm ceil}$ are smaller at
higher redshift.

Thus, given a smaller $\sigma$, we keep more haloes with large $V_{\rm peak}$
that can be scattered to a certain range of $V_{\rm peak}^{\rm scat}$, leading
to a smaller stellar mass incompleteness (see Section 4.1). A smaller $V_{\rm
ceil}$ means to remove fewer haloes with the largest $V_{\rm peak}^{\rm
scat}$, so the sample is more complete at massive-end. This agrees with the
minimum magnitude truncation mentioned in Section 2.1.1, since this threshold
is expected to remove fewer galaxies at higher redshift (Zhai et al., 2017).
To demonestrate this, we plot the completeness–redshift relation in Fig. 9.
The completeness is defined as the ratio of the number of eBOSS LRG targets
with a complete target selection including (Prakash et al., 2016), to that
from a target selection without the $i$-band lower limit $i\geq 19.9$ (Eq.
(1)). Since there are no spectroscopic redshift measurements for objects
excluded by the eBOSS target selection criteria, we match the eBOSS target
catalogues with the DECaLS DR9
data101010https://www.legacysurvey.org/dr9/files (Dey et al., 2019) to make
use of the DECaLS photometric redshift measurements (Zhou et al., 2020) and
78.5 per cent of the eBOSS LRG candidates without the $i$-band lower limit are
found in the DECaLS catalogue.

For the complete samples, i.e., LOWZ and CMASS at $0.43<z<0.51$), we obtain
much tighter constraints on $\sigma$ at different redshifts, as shown in Table
1. Since the $\sigma$ parameter in this study absorbs the incompleteness of
stellar mass function, our measurements provide upper bounds of the intrinsic
scatter between stellar mass and our halo mass proxy $V_{\rm peak}$, i.e.
$\sigma_{\rm int}<0.31$ at $0.2<z<0.33$, $\sigma_{\rm int}<0.36$ at
$0.33<z<0.43$, and $\sigma_{\rm int}<0.46$ at $0.43<z<0.51$ with 95 percent
confidence. We do not consider the measurement at $0.51<z<0.57$ to be robust
as explained in Section 4.1.

###  4.4 Consistency Check with $w_{p}$

In principle, when the 2PCF multipoles of a SHAM catalogue agree well with
those of the corresponding observational data, the $w_{p}$ of the two
catalogues should also be consistent. To investigate the impacts of potential
systematic effects that are usually observed on large scales (Ross et al.,
2016), we measure the $w_{p}$ with two integral constraints for both the best-
fitting SHAM catalogues and observations. At first, we use $w_{p}$ with
$\pi_{\rm max}=80\,h^{-1}\,{\rm Mpc}$ to obtain a good SNR and avoid the
noise-dominated large scales (Mohammad et al., 2020). This $\pi_{\rm max}$
value is a common choice for HOD studies (e.g., Avila et al., 2020b). In this
case, the $w_{p}$ of the SHAM for LOWZ samples agrees with the observed
$w_{p}$, while for CMASS and eBOSS samples, the $w_{p}$ of SHAM catalogues are
generally underestimated compared to those of the observations. We then reduce
$\pi_{\rm max}$ to $30\,h^{-1}\,{\rm Mpc}$, and the $w_{p}$ measured from the
SHAM and observational catalogues become consistent, as shown in Figures
10–12. Note that $\pi_{\rm max}$ values are for both observational and SHAM
$w_{p}$.

Potential uncorrected systematics that affect the 2PCF monopole on $r\gtrsim
80\,h^{-1}\,{\rm Mpc}$ can explain the inconsistency in $w_{p}$ with $\pi_{\rm
max}=80\,h^{-1}\,{\rm Mpc}$. The discrepancy on the large scales of the 2PCF
monopole is observed when comparing observations with the galaxy mocks like
Patchy (Kitaura et al., 2016) and EZmock (Zhao et al., 2021a). Meanwhile,
Huterer et al. (2013) explain that uncorrected photometric systematics can
significantly bias clustering on large scales.

To directly demonstrate the consequences of uncorrected systematics on $w_{p}$
with different $\pi_{\rm max}$, we compare the 2PCF multiples and projected
2PCFs of EZmock mocks without and with observational systematics,
respectively. EZmock mocks with observational systematics are mocks that mimic
the data, including all the systematics mentioned in Section 2.1.3. We do not
apply weight corrections, to investigate the potential biases due to
systematics. The clustering difference is normalized by the quadrature sum of
the standard deviations for EZmock mocks with and without systematics. As is
shown in Fig. 13, the observational systematics is responsible for the
monopole difference, especially at scales larger than $50\,h^{-1}\,{\rm Mpc}$.
Consequently, the bias presented in $w_{p}$ with $\pi_{\rm
max}=80\,h^{-1}\,{\rm Mpc}$ is much larger than that of $w_{p}$ with $\pi_{\rm
max}=20\,h^{-1}\,{\rm Mpc}$. The result reveals the importance of properly
choosing $\pi_{\max}$ to avoid large-scale systematics as much as possible.

Figure 13: The effect of a biased 2PCF monopole on $w_{p}$. The first column
represents the 2PCF multipoles of EZmock mocks without systematics (black
lines with shades) and EZmock mocks with systematics (blue lines with shades).
The second column is the $w_{p}$ comparison between EZmock mocks without
systematics (black lines with shades) and EZmock mocks with systematics (red
lines with shades). The first row is $w_{p}$ with $\pi$ integrating to
80$\,h^{-1}\,{\rm Mpc}$ and the third row is $\pi$ integrating to
20$\,h^{-1}\,{\rm Mpc}$. The second and fourth row for both columns are the
difference between their clustering normalized by the quadrature sum of the
standard deviations for EZmock mocks with and without systematics. The $w_{p}$
difference is suppressed when $\pi_{\rm max}$ equals a smaller value.

##  5 Conclusion

Subhalo Abundance matching is a powerful method of constructing galaxy
catalogue based on high-resolution simulations. We propose a 3-parameter SHAM
algorithm that is more general for different galaxy surveys. Besides the
classical $\sigma$, we introduce $v_{\rm smear}$ that smears the peculiar
velocities of SHAM galaxies to model the effect of the redshift uncertainties,
and $V_{\rm ceil}$ that removes the most massive haloes/galaxies to account
for the completeness of tracers in the massive end due to the target selection
(e.g. luminosity threshold).

We construct SHAM catalogues that reproduce the clustering of BOSS/eBOSS LRGs
on $[5,25]\,h^{-1}\,{\rm Mpc}$ at $0.2<z<1.0$, based on the UNIT simulations.
The 2PCF multipoles and projected 2PCFs of the SHAM catalogues are consistent
with those of BOSS/eBOSS samples, including the entire LOWZ, CMASS and eBOSS
samples, and 9 subsamples in different redshift slices. These results validate
our new SHAM model. However, for some of the samples such as CMASS at
$0.43<z<0.7$ and eBOSS LRG $0.7<z<0.9$, the best-fitting reduced $\chi^{2}$
values for the 2PCF multipoles can be larger than 2. This may be due to
unknown observational systematics. For the bulk CMASS samples, the
inhomogeneity in completeness at different redshift bins can also contribute
to the disagreement.

Since the galaxies at $z<0.6$ are supposed to be complete (Reid et al., 2016),
a SHAM model without $V_{\rm ceil}$ should also work for them. This is
confirmed by the fitting results except for the CMASS sample at $0.51<z<0.57$,
which is rejected at more than $4\sigma$. Therefore, the BOSS LRGs should have
already been incomplete from $z\approx 0.5$. This finding is especially
important for the reconstruction of the stellar mass function.

Using our SHAM method, the 2PCF bias brought by redshift uncertainties can be
quantified through the $v_{\rm smear}$ parameter. Meanwhile, we estimate the
redshift uncertainty statistically with the redshift difference $\Delta v$
obtained from repeat observations. There are two estimators of the statistical
redshift uncertainties: (1) the best-fitting Gaussian dispersions
$\sigma_{\Delta v}$ for the histogram of $\Delta v$; (2) the standard
deviations of $\Delta v$ as $\hat{\sigma}_{\Delta v}$. As expected, both of
them increase monotonically with the effective redshift due to broader
absorption lines at higher redshifts. Both $\sigma_{\Delta v}$ and
$\hat{\sigma}_{\Delta v}$ agree with our CMASS $v_{\rm smear}$ measurements.
For eBOSS samples, $v_{\rm smear}$ agrees with the $\hat{\sigma}_{\Delta v}$
measurements while it is in tension with $\sigma_{\Delta v}$. This can be
explained by the larger outlier fraction of eBOSS histograms. In short, the
standard deviations of the redshift differences are in agreement with $v_{\rm
smear}$ for both the CMASS and eBOSS LRG samples.

However, for the BOSS LOWZ sample, there are significant disagreements between
$v_{\rm smear}$ values and the redshift uncertainties measured from
repetitively observed targets. We have looked into some potential sources of
systematic biases including the robustness of subhaloes from the UNIT
simulations, the SDSS-III spectroscopic pipeline upgrade and the
representativeness of the repeat samples. But none of them can lead to the
observed discrepancy. Despite the fact that we have similar satellite fraction
for LOWZ and CMASS, it is still possible that LOWZ sample is composed of a
different subclass of LRGs that has different satellite properties compared to
LRGs from CMASS and eBOSS. To validate this, we need a more detailed SHAM
model for LOWZ samples. We leave relevant studies to a future work.

We observe the redshift evolution of eBOSS $\sigma$–$V_{\rm ceil}$ posteriors,
indicating the sample is more complete at higher redshift bins. With the
photometric redshift from DECaLS, we confirm the target selection criteria
with a minimum i-band magnitude cut leads to the completeness evolution. Since
the scatter parameter $\sigma$ in our study include both the intrinsic scatter
$\sigma_{\rm int}$ and the incompleteness of galaxy samples, our 2-parameter
SHAM measurements provide the constraints on $\sigma_{\rm int}$ as
$\sigma_{\rm int}<0.31$ for LOWZ at $0.2<z<0.33$, $\sigma_{\rm int}<0.36$ for
LOWZ at $0.33<z<0.43$, and $\sigma_{\rm int}<0.46$ for CMASS at $0.43<z<0.51$.

The projected 2PCFs of the best-fitting SHAM catalogues are consistent with
those of the observations when the integral limit $\pi_{\rm max}$ is as small
as $30\,h^{-1}\,{\rm Mpc}$. However, $w_{p}$ of SHAM catalogues deviates from
the observations when $\pi_{\rm max}=80\,h^{-1}\,{\rm Mpc}$. This can be
explained by potential uncorrected systematics that mainly affect large
scales. A test based on EZmock mocks with and without systematics also proves
that $w_{p}$ evaluated with a $\pi_{\rm max}$ of around $30\,h^{-1}\,{\rm
Mpc}$ is less sensitive to systematic effects, compared to $w_{p}$ with
$\pi_{\rm max}=80\,h^{-1}\,{\rm Mpc}$. The result also supports the choice of
a smaller $\pi_{\rm max}$ when using $w_{p}$ for SHAM and HOD studies (e.g.,
Alam et al., 2020).

To conclude, our 3-parameter SHAM model works well for LRGs at a wide range of
redshift, though there are a few exceptions that may be due to uncorrected
observational systematics. Our algorithm is also effective in estimating
uncertainties of redshift measurements, and detection of sample incompleteness
for most of the SDSS samples. We are going to further improve the SHAM model
by accounting for more properties, such as the satellite fraction. In the
meantime, we also plan to extend our the studies to ELGs. Because their star-
forming processes are quenched in massive haloes (Kauffmann et al., 2004;
Dekel & Birnboim, 2006), the parameter $V_{\rm ceil}$ is foreseen to be
crucial. Later, we shall also construct multi-tracer SHAM models for different
types of tracers with both auto- and cross-correlations for cosmological
analysis.

## Data Availability

The LOWZ and CMASS clustering and random catalogues, Patchy mocks are all from
SDSS data release 12. For eBOSS, the PIP+ANG weighted clusterings can be
provided by FM. The corresponding EZmock mocks can be obtained with request to
CZ. The spAll catalogue used for the redshift difference measurements are
publicly available and the code of the redshift uncertainty measurement can be
obtained upon the request to JB.

## Acknowledgement

JY, CZ and GF acknowledge support from the SNF 200020_175751 “Cosmology with
3D Maps of the Universe” research grant. GR acknowledges support from the
National Research Foundation of Korea (NRF) through Grant No. 2020R1A2C1005655
funded by the Korean Ministry of Education, Science and Technology (MoEST).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

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##  Appendix A The effect of fibre collision

To avoid potential biases of the clustering measurements due to uncorrected
fibre collision effects with nearest-neighbour close-pair weights ($w_{\rm
CP}$), we conduct a new SHAM fitting on the monopole and quadropole in [10,
25]$\,h^{-1}\,{\rm Mpc}$, which are not significantly affected by fibre
collision effects (Guo et al., 2012; Rodríguez-Torres et al., 2016), for LOWZ
at $0.2<z<0.43$. The resulting posterior distributions of parameters are shown
in Fig. 14. As demonstrated in the figure, the constraints are consistent with
those from our fiducial fitting range ([5, 25]$\,h^{-1}\,{\rm Mpc}$) at
1$\sigma$ level. It thus suggests that our SHAM fitting results are robust.

Figure 14: The posteriors of SHAM parameters for LOWZ at $0.2<z<0.43$,
obtained with the fitting of monopole and quadropole at ranges on [10,
25]$\,h^{-1}\,{\rm Mpc}$ (red) and [5, 25]$\,h^{-1}\,{\rm Mpc}$ (blue).

##  Appendix B Blue Samples in LOWZ

In order to indirectly prove the representative of the repeat observation, we
choose to study the fraction of blue galaxies, i.e., $f_{\rm blue}$ in the
repeat measurement catalogue and the galaxy clustering catalogue. The
differentiation of the blue and red samples is achieved by applying an ad hoc
criterion on the colour–redshift diagram. For example, with a constant colour
cut $(g-i)=2.35$, Favole et al. (2016b) selects the blue tail of CMASS samples
and find out that they have different clustering properties from the red
samples. The magnitudes for colour calculation are the Composite Model
Magnitudes (i.e., CMODELMAG; Reid et al., 2016) from the spAll catalogue. We
start with using the same cut on both clustering catalogue and repeated
samples for LOWZ and CMASS (Fig. 15 and Fig. 16). The results in Table 2 shows
that $f_{\rm blue}$ in clustering samples are quite close to $f_{\rm blue}$ of
the repeat samples. Despite of the minor difference, for LOWZ, $f_{\rm blue}$
of the clustering catalogue is always larger than $f_{\rm blue}$ of the
repeated samples, while for CMASS it is the opposite. Additionally, there is
no significant difference in the $\hat{\sigma}_{\Delta v}$ of blue galaxies
and red galaxies that can lead to the disagreement between LOWZ $v_{\rm
smear}$ and LOWZ $\Delta v$.

Figure 15: The colour–redshift diagram for CMASS at $0.43<z<0.7$ and the blue
galaxy threshold (white dashed lines). Left: the colour distribution of
galaxies from the clustering catalogue. Right: the colour distribution of
galaxies from the repeat observation.  Figure 16: The same as Fig. 15 for LOWZ
with two cuts. The first cut is $(g-i)=2.35$ (the white solid line) and the
second one is $(g-i)=2.8z+1.2$ (the white dashed line).

As shown in Fig. 16, a constant cut cannot help to select the blue tail of
LOWZ. A cut varies with the redshift is a better choice. So we apply
$(g-i)=2.8z+1.2$, and find $f_{\rm blue}$ drops to the same level as CMASS
with a constant cut. The clustering catalogue still has $f_{\rm blue}$ larger
than that of the repeat observations and we have small difference between the
red and blue $\hat{\sigma}_{\Delta v}$ just like our findings in the results
of the constant cut. The results can be found in Table 3. So it proves that
the redshift uncertainties measured based on repeat samples should be
representative for the clustering sample.

z range | $f_{\rm blue}$ | $f_{\rm blue}$ | red $\hat{\sigma}_{\Delta v}$ | blue $\hat{\sigma}_{\Delta v}$
---|---|---|---|---
| clst. (%) | repeat (%) | $({\rm s}^{-1}\,{\rm km})$ | $({\rm s}^{-1}\,{\rm km})$
$0.2<z<0.33$ | 88.9 | 87.3 | 29.3$\pm$3.4 | 26.0$\pm$2.5
$0.33<z<0.43$ | 46.1 | 44.6 | 33.1$\pm$1.7 | 33.1$\pm$3.6
$0.2<z<0.43$ | 67.3 | 64.6 | 32.5$\pm$1.5 | 28.8$\pm$2.2
$0.43<z<0.51$ | 37.3 | 37.5 | 46$\pm$2.7 | 48.3$\pm$4.7
$0.51<z<0.57$ | 36.3 | 37.4 | 52.8$\pm$3.4 | 52.6$\pm$3.0
$0.57<z<0.7$ | 37.0 | 38.8 | 61.0$\pm$3.6 | 60.7$\pm$4.8
$0.43<z<0.7$ | 36.9 | 38.0 | 54.0$\pm$1.9 | 54.6$\pm$2.8
Table 2: The $f_{\rm blue}$ for the clustering catalogue and the repeat observation, and the $\hat{\sigma}_{\Delta v}$ of the blue and the red galaxies for LOWZ and CMASS samples at different redshift bins using the constant cut $(g-i)=2.35$. The $f_{\rm blue}$ values of LOWZ clustering catalogue are larger than those of the repeat catalogue, while CMASS samples shows the opposite relation. The difference between the red and blue $\hat{\sigma}_{\Delta v}$ is not significant for LOWZ and CMASS. z range | $f_{\rm blue}$ | $f_{\rm blue}$ | red $\hat{\sigma}_{\Delta v}$ | blue $\hat{\sigma}_{\Delta v}$
---|---|---|---|---
| clst. (%) | repeat (%) | $({\rm s}^{-1}\,{\rm km})$ | $({\rm s}^{-1}\,{\rm km})$
$0.2<z<0.33$ | 30.9 | 29.1 | 26.8$\pm$2.9 | 25.7$\pm$1.7
$0.33<z<0.43$ | 30.5 | 30.2 | 32.7$\pm$1.4 | 34.1$\pm$5.1
$0.2<z<0.43$ | 30.7 | 29.7 | 30.0$\pm$1.5 | 30.5$\pm$3.2
Table 3: The same as Table 2 for LOWZ using the redshift cut $(g-i)=2.8z+1.2$.
The $f_{\rm blue}$s of clustering are still larger than those of the repeat
catalogue and the difference between the red and blue $\hat{\sigma}_{\Delta
v}$ is also small.

##  Appendix C Posteriors of BOSS and eBOSS SHAM

The posteriors of all the 3-parameter SHAM results are presented in Figures
17–22. The top panel of each column shows the marginalized posterior of the
parameter written at the bottom. The remaining three panels are the 2D
posterior contours. Parameters with the maximum likelihood determined by
pyMultinest111111https://github.com/JohannesBuchner/PyMultiNest (Buchner et
al., 2014) are marked in the figures. All the posteriors are from converged
Monte-Carlo chains of Multinest. The multi-modal 1D posteriors of $\sigma$ and
$V_{\rm ceil}$ may be due to the noisy data.

We also present the posteriors of the 2-parameter SHAM without $V_{\rm ceil}$
in Figures 23–25, comparing with those of the corresponding 3-parameter SHAM.
We keep the posteriors of $V_{\rm ceil}$ for the convenience of visualizing
the $V_{\rm ceil}$–$v_{\rm smear}$ degeneracy.

Figure 17: The posteriors of SHAM parameters for LOWZ samples at $0.2<z<0.43$.
The black dots in the contours indicate the position of the best-fitting
parameters. Figure 18: The same as Fig. 17 for LOWZ SHAM at $0.2<z<0.33$
(magenta) and $0.33<z<0.43$ (blue). Figure 19: The same as Fig. 17 for CMASS
SHAM at $0.43<z<0.7$. Figure 20: The same as Fig. 17 for LOWZ SHAM at
$0.43<z<0.51$ (magenta), $0.51<z<0.57$ (blue) and $0.57<z<0.7$ (orange).
Figure 21: The same as Fig. 17 for eBOSS LRG SHAM at $0.6<z<1.0$. Figure 22:
The same as Fig. 17 for eBOSS LRG SHAM at $0.6<z<0.7$ (magenta), $0.6<z<0.8$
(blue), $0.7<z<0.9$ (orange) and $0.8<z<1.0$(red). The evolution of the
$\sigma$–$V_{\rm ceil}$ contour indicates the incompleteness at higher
redshift bins is lower than that of the lower redshift bins as explain in
Section 4.3.  Figure 23: The posteriors of 3-parameter SHAM (magenta) compared
with 2-parameter SHAM (red) for LOWZ samples at $0.2<z<0.43$. Figure 24: The
same as Fig. 23 for LOWZ SHAM comparisons at $0.2<z<0.33$ (magenta, red) and
$0.33<z<0.43$ (cyan, blue). Figure 25: The same as Fig. 23 for CMASS SHAM
comparisons at $0.43<z<0.51$ (magenta, red) and $0.51<z<0.57$ (cyan, blue).