# Construct the emission line galaxy-host halo connection through auto and
cross correlations

Hongyu Gao  Department of Astronomy, School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, People’s Republic of China  Y.P. Jing
Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, People’s Republic of China  Tsung-Dao Lee
Institute, and Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Yun Zheng  Department of Astronomy, School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, People’s Republic of China  Kun Xu
Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, People’s Republic of China

(Received XXX; Revised YYY; Accepted ZZZ)

###### Abstract

We investigate the [O II] emission line galaxy (ELG)-host halo connection via
auto and cross correlations, and propose a concise and effective method to
populate ELGs in dark matter halos without assuming a parameterized halo
occupation distribution (HOD) model. Using the observational data from VIMOS
Public Extragalactic Redshift Survey (VIPERS), we measure the auto and cross
correlation functions between ELGs selected by [O II] luminosity and normal
galaxies selected by stellar mass. Combining the stellar-halo mass relation
(SHMR) derived for the normal galaxies and the fraction of ELGs observed in
the normal galaxy population, we demonstrate that we can establish an accurate
ELG-halo connection. With the ELG-halo connection, we can accurately reproduce
the auto and cross correlation functions of ELGs and normal galaxies both in
real-space and in redshift-space, once the satellite fraction is properly
reduced. Our method provides a novel strategy to generate ELG mock catalogs
for ongoing and upcoming galaxy redshift surveys. We also provide a simple
description for the HOD of ELGs.

Emission line galaxies (459), Redshift surveys (1378), Galaxy dark matter
halos (1880), Dark energy (351), Observational cosmology (1146)

††journal: ApJ††software: Numpy (van der Walt et al., 2011), Scipy (Oliphant,
2007), Matplotlib (Hunter, 2007), Astropy (Astropy Collaboration et al.,
2013), scikit-learn (Pedregosa et al., 2011), emcee (Foreman-Mackey et al.,
2013)

##  1 Introduction

Distinguishing dark energy models from modified gravity theories requires us
to accurately measure the entire evolutionary history of the Universe from the
matter dominance to the dark energy dominance. By analyzing the clustering of
galaxies, we can measure the expansion history and instantaneous expansion
rate of the universe from the baryon acoustic oscillation (BAO, e.g., Cole et
al., 2005; Eisenstein et al., 2005), and the growth rate from the redshift-
space distortion (RSD, e.g., Kaiser, 1987). Sloan Digital Sky Survey (SDSS,
York et al., 2000; Gunn et al., 2006) has performed spectroscopic measurement
for a large number of galaxies in the low-redshift Universe that has been
dominated by dark energy. But for distant galaxies with redshift $z>0.7$,
because their continuum gets faint and most of their optical spectral lines
are redshifted to the infrared band, it becomes more and more difficult to
conduct a large redshift survey at high redshift.

To overcome this difficulty, galaxies with strong [O II] emission lines (ELGs)
have naturally become the main target for next generation redshift surveys
(Newman et al., 2013; Dawson et al., 2016; DESI Collaboration et al., 2016;
Takada et al., 2014). Since the neutral gas can be photo-ionized by the
ultraviolet (UV) radiation of newly formed massive stars and produce [O II]
lines, the main population of [O II] emitters is therefore expected to be
star-forming galaxies. Although the violent nuclear activities caused by super
massive black holes (SMBHs) also have enough energy to ionize oxygen atoms,
the fraction of active galactic nuclei (AGN) in the [O II] emitters is small
(e.g., Comparat et al., 2013). Compared to other spectral features (such as
H$\alpha$, [O III] and $4000\,\rm{\AA}$ break), the main advantage of the [O
II] line is the doublet feature at the wavelengths $\lambda
3727,3729\,\rm{\AA}$, and it can be detected in the optical window up to
redshift 1.6. For example, the Deep Extragalactic Evolutionary Probe 2 (DEEP2,
Newman et al., 2013) has measured the spectrum for more than 50,000 galaxies
at $z\sim 1$, in which the number of [O II] ELGs is dominant. Currently, the
Dark Energy Spectroscopic Instrument (DESI, DESI Collaboration et al., 2016)
is conducting spectroscopic observations for more than 17 million [O II] ELGs
within $0.6<z<1.6$ covering 14,000 square degrees, which makes up for the
vacancy of the luminous red galaxy (LRG) sample at $z>1$. With the help of its
near-infrared spectrometers and large-aperture, the Subaru Prime Focus
Spectrograph (PFS, Takada et al., 2014) will conduct spectroscopic
observations of [O II] ELGs all the way to $z=2.4$. The combination of these
two redshift surveys will increase the number and coverage of observed [O II]
ELGs to an unprecedented level.

Before we can extract the cosmological information from the clustering signal
of [O II] ELGs, we need to understand the connection between these galaxies
and the underlying dark matter. This is also a prerequisite for generating
realistic [O II] ELG mock catalogs (e.g., Osato et al., 2021) for these
redshift surveys. Halo occupation distribution (HOD) has become one of the
most common ways to construct the halo-galaxy connection (e.g., Jing et al.,
1998; Peacock & Smith, 2000; Ma & Fry, 2000; Seljak, 2000; Berlind & Weinberg,
2002; Zheng et al., 2005, 2007; Zehavi et al., 2011; Zu & Mandelbaum, 2015,
2016, 2018; Guo et al., 2016; Rodríguez-Torres et al., 2016; Xu et al., 2016,
2018; Yuan et al., 2018; Wang et al., 2019). Under the HOD framework, the mean
occupation number $\left\langle N\left(M\right)\right\rangle$ of a given
galaxy population in a halo is determined by the halo mass. In addition to
HOD, the statistics related to the physical properties (such as stellar mass
and luminosity) of the galaxies inhabiting a halo of given mass can be
described by the conditional luminosity (stellar mass) function (Yang et al.,
2003; Cooray, 2006; van den Bosch et al., 2007; Yang et al., 2009, 2012;
Rodríguez-Puebla et al., 2015; Guo et al., 2018). Since subhalos can be more
accurately resolved as cosmological simulations improves, the (sub)halo
abundance matching (AM) method (e.g., Wechsler et al., 1998; Wang et al.,
2006; Vale & Ostriker, 2006; Behroozi et al., 2010; Wang & Jing, 2010; Guo et
al., 2010; Simha et al., 2012; Moster et al., 2013; Guo & White, 2014; Guo et
al., 2016; Chaves-Montero et al., 2016; Wechsler & Tinker, 2018; Behroozi et
al., 2019; Xu et al., 2021; Xu & Jing, 2021) has become a more effective way
to link the observable physical quantity (e.g., stellar mass, luminosity) of a
galaxy to its host (sub)halo properties (e.g., halo mass, maximum circular
velocity). For normal galaxies in a stellar mass-selected galaxy sample, their
galaxy-halo connection is relatively easy to understand, because the
monotonically increasing stellar-halo mass relation (SHMR) indicates that the
massive galaxies tend to occupy massive haloes, although it might be affected
by other properties beyond stellar mass due to galaxy assembly bias (e.g.,
Cooper et al., 2010; Wang et al., 2013; Zentner et al., 2014; Hearin et al.,
2015; Mandelbaum et al., 2016; Xu & Zheng, 2020; Hadzhiyska et al., 2020,
2021a; Zu et al., 2021a, b; Cui et al., 2021; Xu & Jing, 2021). However, the
situation may become more complicated for ELGs. Since the quench fraction of
massive galaxies is relatively higher, galaxies with strong emission lines are
not necessarily hosted by massive haloes. On the contrary, low-mass galaxies
are more likely to have strong star formation processes. Therefore, the
probability that a halo hosts an ELG is expected to peak at low-mass, and then
gradually decreases toward the high-mass end (Geach et al., 2012; Contreras et
al., 2013).

Recently, a handful studies have been devoted to studying the [O II] ELG-halo
connection in observations (Favole et al., 2016, 2017; Guo et al., 2019; Avila
et al., 2020; Okumura et al., 2021). For instance, by simultaneously
constraining the SHMR, completeness and quench fraction of the [O II] ELG
sample from the extended Baryon Oscillation Spectroscopic Survey (eBOSS,
Dawson et al., 2016), Guo et al. (2019) found that the typical host halo mass
of eBOSS ELGs is $\sim 10^{12}\,M_{\odot}$ and the satellite fraction is
13%-17%, although the results slightly depend on the assumptions of their
quenched fraction model. They also showed that the completeness of eBOSS ELGs
is less than 10%, which indicates that only galaxies with the strongest [O II]
emissions are selected by eBOSS. Using the [O II] ELGs sample identified by
the narrow-band (NB) filters at $z>1$ in the Subaru Hyper-Suprime Cam (HSC)
survey, Okumura et al. (2021) found that their angular correlation function
can be well fitted by the HOD model proposed by Geach et al. (2012), but the
constraints of the model parameters are poor due to the limited data and the
large parameter space. From the number densities of the HSC NB ELGs ($\sim
10^{-3}\,{\rm Mpc}^{-3}\,h^{3}$) and eBOSS ELGs ($\sim 10^{-4}\,{\rm
Mpc}^{-3}\,h^{3}$), we can easily see that these ELGs are different
populations of [O II] emitters. It is important to study how the ELG-halo
connection depends on the [O II] luminosity.

Different from previous works, we aim to investigate the ELG-halo connection
for different [O II] luminosities by utilizing the cross correlations between
ELGs and normal galaxies. Since the host haloes of ELGs are widely distributed
in mass as we will show, the auto correlation of ELGs actually mixes the
clustering signal of haloes with different mass and is therefore difficult to
interpret. But for the normal galaxies selected by stellar mass, we already
have a relatively clear understanding of their clustering and host haloes
properties. Furthermore, the number density of normal galaxies is higher,
which makes the cross correlation a better measured quantity. Therefore, the
cross correlation of an ELG sample with a stellar mass-selected galaxy sample
will tell us how ELGs are distributed around halos which are derived from the
SHMR of normal galaxies. In this work, we use the galaxy catalog from the
VIMOS Public Extragalactic Redshift Survey (VIPERS111http://vipers.inaf.it,
Guzzo et al., 2014; Garilli et al., 2014; Scodeggio et al., 2018). Unlike the
eBOSS which only selects ELGs with strong emission lines, VIPERS is an
$i$-band limit survey and thus contains ELGs with more moderate [O II]
luminosity that are also the main targets of DESI and PFS. Different from the
HOD modelings mentioned before, we make full use of the ELG-stellar mass
relation in observation without establishing a parameterized model. We will
demonstrate that by randomly assigning ELGs to dark matter halos according to
the SHMR of normal galaxies, we can well repeat the auto and cross correlation
functions in both real-space and redshift-space as long as the satellite
fraction is reduced. The method is simple but effective, which could be a
starting point for constructing the ELG-halo connection for surveys such as
DESI and PFS, and become a test-bed for further improving the connection.

The layout of this paper is organized as follows. In Section 2, we describe
the observational data and numerical simulation used in this work. In Section
3, we introduce our methods to account for the survey selection effects and to
measure correlation functions. The SHMR is derived by the AM method in Section
4. The main results of the ELG-halo connection are presented in Section 5.
Eventually, we give a brief conclusion in Section 6. Unless otherwise stated,
the cosmological parameters used in this paper are:
$\Omega_{\mathrm{m},0}=0.268$, $\Omega_{\Lambda,0}=0.732$ and
$H_{0}=100h\,\mathrm{km\,s^{-1}\,Mpc^{-1}}=71\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$.

Figure 1: The stellar mass and [O II] luminosity distributions of galaxies in the VIPERS sample. The blue points denote the full galaxy sample while the green points represent the subsamples, whose boundaries are plotted as orange dashed lines.  Table 1: The details of four stellar mass-selected subsamples.  Name | Redshift Range | $\log M_{\ast}\,[M_{\odot}]$ | $N_{\mathrm{g}}$
---|---|---|---
$M0$ | $0.5<z\leq 0.8$ | $\left[9.9,10.2\right]$ | 7829
$M1$ | $0.5<z\leq 0.8$ | $\left[10.2,10.5\right]$ | 8355
$M2$ | $0.5<z\leq 0.8$ | $\left[10.5,10.9\right]$ | 8040
$M3$ | $0.5<z\leq 0.8$ | $\left[10.9,\infty\right]$ | 1964
Table 2: The details of four [O II] luminosity-selected subsamples.  Name | Redshift Range | $\log L_{\mathrm{[O\,II]}}\left(z=0.5\right)\,\mathrm{[erg/s]}$ | $N_{\mathrm{g}}$
---|---|---|---
$L0$ | $0.5<z\leq 0.8$ | $\left[40.85,41.15\right]$ | 9349
$L1$ | $0.5<z\leq 0.8$ | $\left[41.15,41.45\right]$ | 11721
$L2$ | $0.5<z\leq 0.8$ | $\left[41.45,41.75\right]$ | 6281
$L3$ | $0.5<z\leq 0.8$ | $\left[41.75,\infty\right]$ | 1693

##  2 Galaxy sample and simulation data

We describe the basics of VIPERS and the properties of our [O II] luminosity-
selected and stellar mass-selected subsamples. The N-body cosmological
simulation used in this study is also introduced in this Section.

###  2.1 VIPERS sample

We use the galaxy catalog of the final public release (PDR-2) (Scodeggio et
al., 2018) of the VIPERS. This survey overlaps two sky fields W1 and W4 of the
Canada-France-Hawaii Telescope Legacy Survey Wide (CFHTLS-
Wide222http://www.cfht.hawaii.edu/Science/CFHLS/), covering about 24 square
degrees. The multi-band magnitudes ($u,g,r,i,z$) of the parent photometric
catalog come from the CFHTLS
T0005333http://www.cfht.hawaii.edu/Science/CFHLS/T0005/. Ancillary photometric
data is supplemented by the VIPERS Multi-Lambda Survey (Moutard et al., 2016),
which matched the CFHTLS T0005 catalog with GALEX (Martin et al., 2005) and
the VISTA Deep Extragalactic Observations (Jarvis et al., 2013), and provided
extra photometry in NUV, FUV and $K_{\mathrm{s}}$ $(K_{\mathrm{video}})$
bands. Galaxies with $i_{\mathrm{AB}}<22.5$ in the parent catalog satisfying
the following color-color criteria

$\displaystyle\left(r-i\right)>0.5\times\left(u-g\right)\quad\mathrm{OR}\quad\left(r-i\right)>0.7$
(1)

are selected as the spectroscopic targets. The spectra of about 90,000
galaxies were measured with the VIMOS multi-object spectrograph on the ESO
Very Large Telescopes (Le Fèvre et al., 2003). Finally, we only include the
VIPERS main galaxy targets ($\mathrm{classFlag}=1$) with high-quality redshift
measurements ($\mathrm{zflag}\geq 2$) in our research.

###  2.2 galaxy subsamples

For the purpose of analyzing the cross correlations of emission line galaxies
and normal galaxies, we divide the galaxy sample in the redshift range
$0.5<z\leq 0.8$ into four [O II] luminosity $L_{\mathrm{[O\,II]}}$-selected
($L0$, $L1$, $L2$ and $L3$) and four stellar mass $M_{\ast}$-selected ($M0$,
$M1$, $M2$ and $M3$) subsamples. After subtracting the continuum, the [O II]
fluxes are measured by fitting a single Gaussian model to the spectrum. The
Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963) is adopted to
derive the best-fitting [O II] fluxes and their uncertainties. We take the
multi-band photometry from VIPERS Multi-Lambda Survey (Moutard et al., 2016)
to model the spectral energy distribution (SED) of galaxies. The LE PHARE
(Arnouts et al., 2002; Ilbert et al., 2006) code is used to perform the SED
fitting and estimate the physical properties (including stellar mass) of
galaxies. More details about the $L_{\mathrm{[O\,II]}}$ measurements and SED
template settings can be found in Gao & Jing (2021)444The cosmological
parameters used in the [O II] luminosity computation and SED fitting process
are $\Omega_{\Lambda,0}=0.7$, $\Omega_{m,0}=0.3$ and
$H_{0}=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, which are slightly different from
what we adopted in this study. But this does not affect our subsequent
analysis, because we mainly care about the relative difference between
different subsamples rather than their absolute value..

The mass and [O II] luminosity distributions of the full sample are displayed
as blue points in Figure 1, in which the four $M_{\ast}$-selected and
$L_{\mathrm{[O\,II]}}$-selected subsamples are also shown as green points. We
present more details of each subsample in Table 1 and 2.

Since the stellar mass function (SMF) evolves relatively weakly at $z<1$
(Pozzetti et al., 2007, 2010; Davidzon et al., 2013) , we apply flat stellar
mass cuts to construct $M_{\ast}$-selected subsamples. In order to determine
the stellar mass completeness limit of the galaxy sample, we follow the same
technique proposed by Pozzetti et al. (2010) (see also Davidzon et al.
(2013)). The 90% stellar mass completeness limit at $z\sim 0.6$ is
$10^{9.8}\,M_{\odot}$, so we take a lower boundary
$M_{\ast}=10^{9.9}\,M_{\odot}$ for the first subsample $M0$.

As for the $L_{\mathrm{[O\,II]}}$-selected subsamples, we adopt redshift
evolution cuts to account for the evolution of [O II] luminosity function.
Referring to the parameterized evolution model of the characteristic
luminosity
$L_{\mathrm{[O\,II]},\star}\left(z\right)=L_{\mathrm{[O\,II]},\star}\left(0\right)\left(1+z\right)^{\beta_{L}}$
provided by Comparat et al. (2016), we define the $L_{\mathrm{[O\,II]}}$ cut
at $z$ as

$\displaystyle\log L^{\mathrm{cut}}_{\mathrm{[O\,II]}}\left(z\right)=\log
L^{\mathrm{cut}}_{\mathrm{[O\,II]}}\left(z=0.5\right)+\log\left(\frac{1+z}{1+0.5}\right)^{\beta_{L}}$
(2)

with $\beta_{L}=2.33$. To ensure the completeness of [O II] detection, we set
the lower boundary of the first subsample $L0$ as
$L_{\mathrm{[O\,II]}}\left(z=0.5\right)=10^{40.85}\,\mathrm{erg\,s^{-1}}$,
which is above the $5\sigma$ detection limit
($F^{\mathrm{lim}}_{\mathrm{[O\,II]}}=3.5\times
10^{-17}\,\mathrm{erg\,s^{-1}\,cm^{-2}}$) of the VIMOS with an exposure time
of 0.75 hours (Comparat et al., 2016).

Figure 2: The redshift distributions of the galaxy and random samples. The
blue histogram shows the full galaxy sample (W1 and W4 fields) of VIPERS. The
distribution of the galaxy sample weighted by $w^{\mathrm{SSR}}\times
w^{\mathrm{TSR}}$ is displayed as the green histogram, which shows a slightly
difference from the un-weighted one. The red solid line represents the radial
distribution of the random sample generated with the $V_{\mathrm{max}}$
technique as described in the text. The numbers of weighted galaxies and
random points have been properly normalized.

###  2.3 Simulation

One of the high resolution N-body cosmological simulations of CosmicGrowth
(Jing, 2019) is used to study the galaxy-halo connection. This simulation is
performed by the $\mathrm{P^{3}M}$ method (Jing & Suto, 2002) and has
$3072^{3}$ dark matter particles in a $600\,\mathrm{Mpc}\,h^{-1}$ box with the
standard $\Lambda$CDM cosmological parameters: $\Omega_{\mathrm{m}}=0.268$,
$\Omega_{\Lambda}=0.732$, $h=0.71$, $n_{\mathrm{s}}=0.968$ and
$\sigma_{8}=0.83$. The haloes and subhaloes are identified by the friends-of-
friends algorithm (FOF) (Davis et al., 1985) and the Hierarchical-Bound-
Tracing algorithm (HBT+) (Han et al., 2012, 2018), respectively.

Although the [O II] ELGs are considered to be more likely to exist in low-mass
haloes ($\sim 10^{12}\,M_{\odot}$) (e.g., Favole et al., 2016; Guo et al.,
2019; Hadzhiyska et al., 2021b; Okumura et al., 2021), the mass resolution
$m_{\mathrm{p}}=5.54\times 10^{8}\,M_{\odot}\,h^{-1}$ of particles in our
simulation is sufficient to resolve them. The halo mass $M_{\mathrm{h}}$ is
defined as its viral mass $M_{\mathrm{vir}}$ that is the mass enclosed by a
sphere with an average density of $\Delta_{\mathrm{vir}}\left(z\right)$ times
the critical density of the universe (Gunn & Gott, 1972; Bryan & Norman,
1998). The subhalo accretion mass $M_{\mathrm{s}}$ is defined as its virial
mass at the last snapshot before merging into the current host halo. In
addition, we have carefully dealt with the small subhalos that have been
almost or even completely stripped by the tidal force. Using the fitting
formula proposed by Jiang et al. (2008), we trace the merger history of those
subhalos with less than 20 particles and calculate their merger time scale to
judge whether they can remain distinct as subhalos. Finally, the snapshot with
$z=0.663$, which is close to the mean redshift of our galaxy sample, is chosen
for our analysis. To make a fair comparison with the observations, we have
incorporated the RSD effects to these simulated haloes (subhaloes). We choose
the $z$-axis as the line of sight and define the redshift of the center of the
simulation box as $0.663$. For a halo (subhalo), the cosmological redshift
$z_{\mathrm{c}}$ is given according to its comoving distance to the center of
the box, while the redshift $z_{\mathrm{p}}$ caused by peculiar motion is
calculated by $v_{z}/c$, where $v_{z}$ is the velocity in z-direction and $c$
is the speed of light. Besides, we add to $v_{z}$ a velocity randomly derived
from a Gaussian distribution of the dispersion $\sigma_{v}=c\sigma_{z}$, where
$\sigma_{z}=0.00054$ is the typical redshift uncertainty for VIPERS (Scodeggio
et al., 2018). Finally, the updated $z$-axis coordinate of a halo (subhalo) is
converted from its observed redshift
$z_{\mathrm{obs}}=\left(1+z_{\mathrm{c}}\right)\left(1+z_{\mathrm{p}}\right)-1$.

##  3 Measurement of galaxy clustering

In this Section, we carefully correct the selection effects in VIPERS and
measure the cross (auto) correlation functions for different galaxy subsamples
both in redshift-space and real-space.

###  3.1 Selection functions

In order to achieve an accurate measurement of galaxy clustering, we should
understand and correct for the selection functions listed below.

  1. 1.

The survey masks in VIPERS. We can account for these survey masks by applying
the same sky geometry to the random sample.

  2. 2.

Target sampling rate (TSR). Some galaxies in the parent photometric catalog
cannot be spectroscopically observed due to the limited number of slits. This
effect can bias the targeting of galaxies in the dense region due to the
uniform distribution of the slits, and the clustering of galaxies is
underestimated. This effect can be corrected by up-weighting
$w^{\mathrm{TSR}}=1/\mathrm{TSR}$ (de la Torre et al., 2013; Scodeggio et al.,
2018) for each galaxy.

  3. 3.

Spectroscopic success rate (SSR). It quantifies the probability that the
redshift of a galaxy targeted by the VIMOS can be successfully measured (i.e.
$\mathrm{zflag}\geq 2$). By exploring the dependence of SSR on the multi-
dimensional parameter space, Scodeggio et al. (2018) evaluates the SSR for
each galaxy based on a nearest-neighbor algorithm. Therefore we also up-weight
each galaxy with $w^{\mathrm{SSR}}=1/\mathrm{SSR}$.

  4. 4.

Slit collisions. Similar to fiber collision, if the distance between the two
galaxies is less than the physical size of the silt, only the spectrum of one
galaxy can be observed. Additionally, in order to avoid the overlap of spectra
in the VIMOS detectors, the spectra of two galaxies with distance below a
specific size along the direction perpendicular to silt cannot be observed at
the same time. The combination of these two effects will suppress the
clustering of galaxies at small scales. Follow the method of Pezzotta et al.
(2017), we calculate the angular weights $w^{\mathrm{A}}\left(\theta\right)$
using 153 VIPERS mock samples (see APPENDIX A) to correct the slit collision
effect for galaxy pairs.

  5. 5.

$i$-band magnitude limit and color sampling rate (CSR). VIPERS adopts the
$i^{\mathrm{AB}}<22.5$ flux cut and a color-color cut (Equation 1) to
construct a flux-limited sample at $z>0.5$, which introduces two radial
selection functions to the redshift distribution of galaxies. Guzzo et al.
(2014) provides an accurate model
$\mathrm{CSR}\left(z\right)=1/2-1/2\operatorname{erf}\left[b\left(z_{\mathrm{t}}-z\right)\right]$
with $b=10.8$ and $z_{\mathrm{t}}=0.444$ for the radial weight
$w^{\mathrm{CSR}}=1/\mathrm{CSR}$ to describe the completeness of the color-
color selection. To account for the combination of these two radial selection
effects, we use the $V_{\mathrm{max}}$ method (Cole, 2011; de la Torre et al.,
2013; Pezzotta et al., 2017; de la Torre et al., 2017; Yang et al., 2020) to
generate a smooth redshift distribution for the random sample. We present the
detail of the $V_{\mathrm{max}}$ method in APPENDIX B. In Figure 2, we can see
that the redshift distribution of the random sample thus generated is well
consistent with the observed one.

After considering these selection effects, we can estimate the completeness-
corrected number density of our galaxy subsamples through

$\displaystyle
n_{\mathrm{g}}=\sum_{i}^{N_{\mathrm{g}}}\frac{w^{\mathrm{TSR}}_{i}w^{\mathrm{SSR}}_{i}w^{\mathrm{CSR}}_{i}}{V_{\mathrm{max},i}},$
(3)

where $V_{\mathrm{max},i}$ is computed with

$\displaystyle
V_{\mathrm{max},i}=\frac{A_{\mathrm{eff}}}{3\times\left(180/\pi\right)^{2}}\times\left[D^{3}_{\mathrm{com}}\left(z_{\mathrm{max},i}\right)-D^{3}_{\mathrm{com}}\left(0.5\right)\right],$
(4)

where $A_{\mathrm{eff}}=16.324\,\mathrm{deg^{2}}$ is the effective sky area of
VIPERS and $D_{\mathrm{com}}$ is the comoving distance.

###  3.2 Estimation of correlation function

To measure the galaxy clustering in redshift-space, we decompose the
separation vector $\boldsymbol{s}=\boldsymbol{s_{1}}-\boldsymbol{s_{2}}$ of
two galaxies into two components $r_{\pi}$ and $r_{\mathrm{p}}$. $r_{\pi}$ can
be obtained by projecting $\boldsymbol{s}$ along the line-of-sight,
$r_{\pi}=\left(\boldsymbol{s}\cdot\boldsymbol{l}\right)/\left|\boldsymbol{l}\right|$
with $\boldsymbol{l}=\left(\boldsymbol{s_{1}}+\boldsymbol{s_{2}}\right)/2$,
and $r_{\mathrm{p}}$ is calculated as $\sqrt{s^{2}-r^{2}_{\pi}}$. We choose
twelve $r_{\mathrm{p}}$ bins from $0.12$ to $30\,\mathrm{Mpc}\,h^{-1}$ with an
equal logarithmic interval and forty $r_{\pi}$ bins from $0$ to
$40\,\mathrm{Mpc}\,h^{-1}$ with an equal linear interval. The redshift-space
cross (auto) correlation functions for different galaxy subsamples are
measured utilizing the Landy-Szalay estimator (Landy & Szalay, 1993; Szapudi &
Szalay, 1998)

$\displaystyle\xi_{xy}\left(r_{\mathrm{p}},r_{\mathrm{\pi}}\right)=\left[\frac{D_{x}D_{y}-D_{x}R_{y}-D_{y}R_{x}+R_{x}R_{y}}{R_{x}R_{y}}\right],$
(5)

where $x$, $y$ indicate different samples ($x=y$ for the auto correlation).
The normalized pair counts for galaxy-galaxy, galaxy-random and random-random
are calculated by

$\displaystyle D_{x}D_{y}\left(r_{\mathrm{p}},r_{\pi}\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{N_{\mathrm{g},x}}\sum_{j=1}^{N_{\mathrm{g},y}}w^{\mathrm{A}}\left(\theta_{ij}\right)w^{\mathrm{c}}_{i}w^{\mathrm{c}}_{j}\Theta_{ij}\left(r_{\mathrm{p}},r_{\pi}\right)}{\sum_{i=1}^{N_{\mathrm{g},x}}\sum_{j=1}^{N_{\mathrm{g},y}}w^{\mathrm{A}}\left(\theta_{ij}\right)w^{\mathrm{c}}_{i}w^{\mathrm{c}}_{j}}$
$\displaystyle D_{x}R_{y}\left(r_{\mathrm{p}},r_{\pi}\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{N_{\mathrm{g},x}}\sum_{j=1}^{N_{\mathrm{r},y}}w^{\mathrm{c}}_{i}\Theta_{ij}\left(r_{\mathrm{p}},r_{\pi}\right)}{N_{\mathrm{r},y}\sum_{i=1}^{N_{\mathrm{g},x}}w^{\mathrm{c}}_{i}}$
$\displaystyle D_{y}R_{x}\left(r_{\mathrm{p}},r_{\pi}\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{N_{\mathrm{g},y}}\sum_{j=1}^{N_{\mathrm{r},x}}w^{\mathrm{c}}_{i}\Theta_{ij}\left(r_{\mathrm{p}},r_{\pi}\right)}{N_{\mathrm{r},x}\sum_{i=1}^{N_{\mathrm{g,y}}}w^{\mathrm{c}}_{i}}$
$\displaystyle R_{x}R_{y}\left(r_{\mathrm{p}},r_{\pi}\right)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{N_{\mathrm{r},x}}\sum_{j=1}^{N_{\mathrm{r},y}}\Theta_{ij}\left(r_{\mathrm{p}},r_{\pi}\right)}{N_{\mathrm{r},x}N_{\mathrm{r},y}},$
(6)

where $\Theta_{ij}$ is equal to 1 only when a galaxy pair falls into this
$\left(r_{\mathrm{p}},r_{\pi}\right)$ bin, and the pair counts have been up-
weighted by $w^{\mathrm{c}}=w^{\mathrm{SSR}}\times w^{\mathrm{TSR}}$ and
$w^{\mathrm{A}}\left(\theta\right)$ as mentioned in Section 3.1.

The $\xi_{xy}\left(r_{\mathrm{p}},r_{\mathrm{\pi}}\right)$ is integrated along
the line-of-sight to give the real-space projected correlation function (Davis
& Peebles, 1983)

$\displaystyle w^{xy}_{\mathrm{p}}\left(r_{\rm p}\right)=2\int_{0}^{r_{\pi,\rm
max}}\xi_{xy}\left(r_{\rm p},r_{\pi}\right)\mathrm{d}r_{\pi},$ (7)

with $r_{\pi,\rm max}=40\,\mathrm{Mpc}\,h^{-1}$. We also employ the same upper
limit of the integration when modeling the $w^{xy}_{\mathrm{p}}\left(r_{\rm
p}\right)$ in the simulation to make a fair comparison.

The covariance matrix of the measured $w^{xy}_{\mathrm{p}}\left(r_{\rm
p}\right)$ is estimated with the jackknife technique. We divide the entire
survey into 24 fields (16 for W1 and 8 for W4) with an area of approximately
one square degree for each field, and the covariance matrix of the measured
$w^{xy}_{\mathrm{p}}\left(r_{\rm p}\right)$ can be estimated with

$\displaystyle\mathrm{Cov}\left(i,j\right)=\frac{N-1}{N}\sum_{k=1}^{N}\left(w^{k}_{\mathrm{p},i}-\bar{w}_{\mathrm{p},i}\right)\left(w^{k}_{\mathrm{p},j}-\bar{w}_{\mathrm{p},j}\right),$
(8)

where $N=24$ is the number of jackknife samples and $i$ ($j$) denotes the $i$
($j$)-th $r_{\mathrm{p}}$ bin.

In analogy to the way of $w_{\mathrm{p}}\left(r_{\rm p}\right)$ in real-space,
we also measure the multiple moments of the correlation functions in redshift-
space. The monopole $\xi_{0}\left(s\right)$, quadrupole
$\xi_{2}\left(s\right)$ and hexadecapole $\xi_{4}\left(s\right)$ (Hamilton,
1992) are defined as

$\displaystyle\xi_{l}\left(s\right)=\frac{2l+1}{2}\int_{-1}^{1}\xi\left(s,\mu\right)L_{l}\left(\mu\right)$
(9)

where $L_{l}\left(\mu\right)$ is the Legendre function, $s$ is binned from
$0.12$ to $30\,\mathrm{Mpc}\,h^{-1}$ with an equal logarithmic interval and
the $\mu$ is binned with a linear width $\Delta\mu=0.1$.

All the measured correlation functions are shown as data points with error
bars in Figures 3, 6, 7 and 9.

Figure 3: The projected cross (auto) correlation functions for the
$M_{\ast}$-selected subsamples, and the SHMR model derived by the AM approach.
In the left four panels, the data points with error bars denote the
observational measurements. The $w_{\mathrm{p}}$ between different subsamples
is marked with different colors. Except for $w^{M0M3}_{\mathrm{p}}$ of
$[M0,M3]$, all observation data are used in the fitting process (see Section
4.3 for details). The best-fitting $w_{\mathrm{p}}$ model as well as its
$1\sigma$ scatter is plotted as solid line with shadow region. The reduced
$\chi^{2}$ is also denoted in the fourth panel. We present the best-fitting
SHMR model in the rightmost panel. The SHMR model derived by Wang & Jing
(2010) is also shown as the blue solid line.  Figure 4: The posterior
distributions of the parameters in the unified SHMR model. The 1-D PDF of each
parameter is plotted as a histogram at the top panel of each column, where the
median value and $1\sigma$ uncertainty is also labeled. The 2-D joint PDF of
each parameter pair is shown as a contour with three confidence levels
($68\%$, $95\%$ and $99\%$).

##  4 Determining the stellar-halo mass relation

Before modeling the cross correlations of ELGs with normal galaxies, we first
establish the connection between stellar mass of normal galaxies and their
halo mass. To derive the SHMR by AM method, we use the model proposed by Wang
& Jing (2010) and adopt an efficient way (Zheng & Guo, 2016; Guo et al., 2016)
to calculate the modeled correlation functions.

###  4.1 The abundance matching model

The conditional probability distribution function (PDF) that a galaxy with
stellar mass $M_{\ast}$ is hosted by a (sub)halo with mass $M$ is assumed to
obey a Gaussian distribution

$\displaystyle
p(M_{\ast}|M)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{\left(\log
M_{\ast}-\log\left\langle
M_{\ast}|M\right\rangle\right)^{2}}{2\sigma^{2}}\right].$ (10)

We adopt the parameterized mean relation $\left\langle
M_{\ast}|M\right\rangle$ proposed by Wang & Jing (2010) (see also Wang et al.
(2006))

$\displaystyle\left\langle
M_{\ast}|M\right\rangle=\frac{2k}{\left(\frac{M}{M_{0}}\right)^{-\alpha}+\left(\frac{M}{M_{0}}\right)^{-\beta}},$
(11)

where $\alpha$ and $\beta$ quantify the slopes of two power-law forms
separated at $M_{0}$, and $k$ is a normalization constant. In principle, the
$p(M_{\ast}|M)$ for central and satellite galaxies should be modeled
separately to account for possible different formation histories. For the
satellites, the current stellar mass depends not only on the accretion mass
but also on the subsequent evolution after infalling (Yang et al., 2012).
However, the difference of the $M_{\ast}$-$M$ relationship between halo and
subhalo is small (Wang & Jing, 2010), and the difference should not be
important given the current sample size of VIPERS (see below). Therefore, we
adopt a unified $M_{\ast}$-$M$ relationship for halos and for subhalos with
the same set of parameters: $\alpha,\beta,M_{0},k,\sigma$.

###  4.2 The tabulated correlation functions

When exploring the parameter space, we usually need to populate haloes
(subhaloes) with galaxies based on updated AM model parameters and calculate
the correlation functions of modeled galaxies by many times. It will consume a
significant amount of CPU time if the correlation functions are not calculated
efficiently. Therefore, we extend the tabulated method (Zheng & Guo, 2016; Guo
et al., 2016) to calculate the cross correlation function in the simulation.
The key of this method is to prepare a table for the correlation functions of
different haloes (subhaloes) binned by mass or other physical properties.
Different weights are assigned to the tabulated correlation functions
according to the AM model, and the combination yields the correlation function
of the modeled galaxies. In this way, the haloes (subhaloes) in our simulation
are divided into 500 tiny mass bins with a width of $\Delta\log M=0.01$
ranging from $10^{10}$ to $10^{15}\,M_{\odot}\,h^{-1}$. The correlation
functions of halo-halo, halo-subhalo and subhalo-subhalo for these bins are
then measured and organized into three tables each with $500\times 500$
elements. Eventually, the modeled $w^{xy}_{\mathrm{p}}$ for two galaxy samples
$x$ and $y$ is computed by

$\displaystyle\begin{aligned} w^{xy}_{\mathrm{p}}(r_{\mathrm{p}})=\\\
\sum_{i,j}\frac{\bar{n}_{\mathrm{h},i}\bar{n}_{\mathrm{h},j}}{\bar{n}_{x}\bar{n}_{y}}&P_{x}\left(M_{\mathrm{h},i}\right)P_{y}\left(M_{\mathrm{h},j}\right)w_{\mathrm{p,hh}}\left(r_{\mathrm{p}}|M_{\mathrm{h},i},M_{\mathrm{h},j}\right)\\\
+\sum_{i,j}\frac{\bar{n}_{\mathrm{h},i}\bar{n}_{\mathrm{s},j}}{\bar{n}_{x}\bar{n}_{y}}[&P_{x}\left(M_{\mathrm{h},i}\right)P_{y}\left(M_{\mathrm{s},j}\right)\\\
+&P_{y}\left(M_{\mathrm{h},i}\right)P_{x}\left(M_{\mathrm{s},j}\right)]w_{\mathrm{p,hs}}\left(r_{\mathrm{p}}|M_{\mathrm{h},i},M_{\mathrm{s},j}\right)\\\
+\sum_{i,j}\frac{\bar{n}_{\mathrm{s},i}\bar{n}_{\mathrm{s},j}}{\bar{n}_{x}\bar{n}_{y}}&P_{x}\left(M_{\mathrm{s},i}\right)P_{y}\left(M_{\mathrm{s},j}\right)w_{\mathrm{p,ss}}\left(r_{\mathrm{p}}|M_{\mathrm{s},i},M_{\mathrm{s},j}\right),\end{aligned}$
(12)

where $i,j$ denote different halo (subhalo) bins. The probabilities that the
central and satellite galaxies in the $M_{\ast}$-selected subsample $x$ are
hosted by the halo with $M_{\mathrm{h},i}$ and subhalo with $M_{\mathrm{s},i}$
are expressed as

$\displaystyle\begin{aligned}
P_{x}\left(M_{\mathrm{h},i}\right)&=P_{\mathrm{cen},x}\left(M^{\mathrm{min}}_{\ast,x}<M_{\ast}<M^{\mathrm{max}}_{\ast,x}|M_{\mathrm{h},i}\right)\\\
&=\int_{M^{\mathrm{min}}_{\ast,x}}^{M^{\mathrm{max}}_{\ast,x}}p\left(M_{\ast}|M_{\mathrm{h},i}\right)\mathrm{d}M_{\ast}\end{aligned}$
(13)

and

$\displaystyle\begin{aligned}
P_{x}\left(M_{\mathrm{s},i}\right)&=P_{\mathrm{sat},x}\left(M^{\mathrm{min}}_{\ast,x}<M_{\ast}<M^{\mathrm{max}}_{\ast,x}|M_{\mathrm{s},i}\right)\\\
&=\int_{M^{\mathrm{min}}_{\ast,x}}^{M^{\mathrm{max}}_{\ast,x}}p\left(M_{\ast}|M_{\mathrm{s},i}\right)\mathrm{d}M_{\ast},\end{aligned}$
(14)

where $M^{\mathrm{min}}_{\ast,x}$ ($M^{\mathrm{max}}_{\ast,x}$) is the lower
(upper) boundary of the $M_{\ast}$-selected subsample $x$ and $p(M_{\ast}|M)$
is the conditional PDF of the stellar mass defined in Equation 10. The number
density of the subsample $x$ can be calculated by

$\displaystyle\bar{n}_{x}=\sum_{i}\left[\bar{n}_{\mathrm{h},i}P_{x}\left(M_{\mathrm{h},i}\right)+\bar{n}_{\mathrm{s},i}P_{x}\left(M_{\mathrm{s},i}\right)\right].$
(15)

###  4.3 Fitting procedure

In the observation, we measure four auto correlation functions
($w^{M0M0}_{\mathrm{p}}$, $w^{M1M1}_{\mathrm{p}}$, $w^{M2M2}_{\mathrm{p}}$ and
$w^{M3M3}_{\mathrm{p}}$), six cross correlation functions
($w^{M0M1}_{\mathrm{p}}$, $w^{M0M2}_{\mathrm{p}}$, $w^{M0M3}_{\mathrm{p}}$,
$w^{M1M2}_{\mathrm{p}}$, $w^{M1M3}_{\mathrm{p}}$, $w^{M2M3}_{\mathrm{p}}$) and
four galaxy number densities ($n_{M0}$, $n_{M1}$, $n_{M2}$ and $n_{M3}$) for
the stellar mass-selected subsamples. The likelihood function is proportional
to $\exp\left(-\chi^{2}/2\right)$, where $\chi^{2}$ is defined as

$\displaystyle\begin{aligned} &\chi^{2}=\\\
&\sum_{i=0}^{3}\sum_{j=i}^{3}\left(w^{MiMj}_{\mathrm{p}}-w^{MiMj}_{\mathrm{p,mod}}\right)^{T}C^{-1}_{MiMj}\left(w^{MiMj}_{\mathrm{p}}-w^{MiMj}_{\mathrm{p,mod}}\right)\\\
&+\sum_{i=0}^{3}\frac{\left(n_{Mi}-n_{Mi,\mathrm{mod}}\right)^{2}}{\sigma^{2}_{Mi}},\end{aligned}$
(16)

where $w^{MiMj}_{\mathrm{p,mod}}$ and $n_{Mi,\mathrm{mod}}$ are the model
predictions. The inverse of the covariance matrix $C^{-1}_{MiMj}$ calculated
by Equation 8 is multiplied by a bias-correction factor
$(N_{\mathrm{jack}}-N_{\mathbb{d}}-2)/(N_{\mathrm{jack}}-1)$ (Hartlap et al.,
2007), where $N_{\mathrm{jack}}=24$ and $N_{\mathbb{d}}=12$ are the number of
jackknife subsamples and $r_{\mathrm{p}}$ bins respectively. Uncertainty of
the galaxy number density $\sigma_{Mi}$ is the field-to-field variation in
different jackknife fields. Particularly, since the red satellite galaxies in
the $M0$ subsample may be slightly incomplete at $z>0.7$, the one-halo term of
the cross correlation between $M0$ and $M3$, which mainly contains the massive
central galaxies, is more likely to be suppressed if the red satellite
galaxies in $M0$ are missing. Conservatively, we remove
$w^{M0M3}_{\mathrm{p}}$ (corresponding to $i=0,j=3$ in Equation 16) in our
fitting. The degree of freedom is therefore ${\rm dof}=12\times 9+4-5=107$. In
Bayesian theory, the posterior distribution is proportional to the likelihood
function times the prior of the parameters. We set wide priors for the five
parameters: $10<\log M_{0}<13$, $0.1<\alpha<0.5$, $1<\beta<5$, $9<\log k<12$
and $0<\sigma<1$. An Markov Chain Monte Carlo (MCMC) analysis is performed
with emcee (Foreman-Mackey et al., 2013). The posterior distributions of the
model parameters are shown in Figure 4. Overall, all the five parameters are
well determined.

We present the best-fitting $w_{\mathrm{p}}$ as well as the SHMR as solid
lines in Figure 3. The reduced $\chi^{2}$ is equal to 1.12, indicating a good
overall fit. Compared to the SHMR model derived by Wang & Jing (2010) at
$z\sim 0.8$ using VVDS observation (Pozzetti et al., 2007), $\log M_{0}$ is
slightly larger and $\alpha$ is slightly smaller in our model ($\log
M_{0}=11.64$ and $\alpha=0.29$ in their unified model). This is partly because
there is a degeneracy of $\log M_{0}$ and $\alpha$, which is also evident in
the $\log M_{0}$-$\alpha$ contour in Figure 4. Nevertheless, with the larger
galaxy sample of VIPERS, we have imposed tighter constraints on the SHMR model
at $z\sim 0.6$.

Figure 5: The fractions of the ELGs in the $L_{\mathrm{[O\,II]}}$-selected
subsamples, in terms of the whole galaxy population, as a function of stellar
mass. The measured $F_{L}\left(M_{\ast}\right)$ as well as its Poisson error
for different subsamples are denoted as the data points with different colors.
The linear interpolation of the observed $F_{L}\left(M_{\ast}\right)$-$\log
M_{\ast}$ relationships are shown as solid lines with corresponding colors.

##  5 Construct the halo occupation of ELGs

In this section, we aim to propose an efficient way to construct the ELG-halo
connection. We investigate how to populate the haloes with ELGs in the
simulation with the measured ELG-stellar mass relation and SHMR. We test our
method in both real-space and redshift-space. We also propose a model for HOD
modeling of ELGs.

###  5.1 ELG-stellar mass relation in the observation

We first measure the fraction of ELGs in the whole population of galaxies as a
function of stellar mass in the observation. For each
$L_{\mathrm{[O\,II]}}$-selected subsample, we divide the galaxies into twenty
$\log M_{\ast}$ bins ranging from $\log M_{\ast}=8$ to $12\,M_{\odot}$ with a
bin width $\Delta\log M_{\ast}=0.2$, and compute the weighted number density
$n\left(L_{\mathrm{[O\,II]}},M_{\ast}\right)$ in each bin using Equation 3.
Then the fraction of each $L_{\mathrm{[O\,II]}}$-selected subsample at a given
stellar mass is defined as

$\displaystyle
F_{L}\left(M_{\ast}\right)=\frac{n\left(L_{\mathrm{[O\,II]}},M_{\ast}\right)}{n\left(M_{\ast}\right)}.$
(17)

where the number density $n\left(M_{\ast}\right)$ of all galaxies for a
stellar mass bin can by estimated by integrating the galaxy stellar mass
function (SMF) $\Phi\left(M_{\ast}\right)$:

$\displaystyle n\left(M_{\ast}\right)=\int_{\log M_{\ast}-\Delta\log
M_{\ast}/2}^{\log M_{\ast}+\Delta\log
M_{\ast}/2}\Phi\left(M_{\ast}\right)\mathrm{d}\log M_{\ast}.$ (18)

Here we adopt the SMF measured by McLeod et al. (2021) in the redshift range
$0.25<z<0.75$. McLeod et al. (2021) has combined the data from the Hubble
Space Telescope (HST) CANDELS fields (Grogin et al., 2011; Koekemoer et al.,
2011) and other ground-based surveys with deep photometric measurements, and
provided the best-fitting parameters of the double Schechter function
(Schechter, 1976).

In Figure 5, we show the fraction $F_{L}\left(M_{\ast}\right)$ for different
$L_{\mathrm{[O\,II]}}$-selected subsamples. The error bars represent the
Poisson errors of the weighted number counts. Firstly, we note that the shapes
of $F_{L}\left(M_{\ast}\right)$ for the four subsamples are similar, while the
locations of the peaks of $F_{L}\left(M_{\ast}\right)$ shift slightly from
$10^{9.3}$ to $10^{9.7}\,M_{\odot}$ with [O II] luminosity increasing.
Furthermore, at the low-mass end, the $i$-band magnitude limit may have led to
a rapid decrease of the number of galaxies, thus causing the rapid drop of
$F_{L}\left(M_{\ast}\right)$. The gradual decrease of
$F_{L}\left(M_{\ast}\right)$ at the high-mass is expected, because more
massive galaxies are more likely to stop their star formation and become
quiescent. Moreover, the $F_{L}\left(M_{\ast}\right)$ of both $L2$ and $L3$
show an upturn at $\log M_{\ast}>11.3\,M_{\odot}$. This feature might imply
that the galaxies at the high-mass end are likely to host AGN, whose violent
activities are sufficient enough to generate strong [O II] emissions (e.g.,
Kocevski et al., 2011). Nevertheless, the AGN contamination cannot
significantly affect our results because the number of the massive galaxies
with $\log M_{\ast}>11.3\,M_{\odot}$ is very small (only 12 galaxies in $L2$
and 10 galaxies in $L3$). Instead of using a parameterized model, we linearly
interpolate the $F_{L}\left(M_{\ast}\right)$-$\log M_{\ast}$ relationships to
preserve the observed information. Moreover, it is worth mentioning that Guo
et al. (2019) points out that the completeness of the ELG sample in eBOSS
varies from 1% to 10% at different stellar masses, which is comparable to the
$F_{L}\left(M_{\ast}\right)$ of the $L2$ or $L3$ subsample. This is due to the
$g$-band magnitude limit of eBOSS (Raichoor et al., 2017), which causes the
majority of the selected ELGs to be luminous ones.

Figure 6: The observed projected cross (auto) correlation functions for the
$L_{\mathrm{[O\,II]}}$-selected subsamples, compared with the AM model
predictions by randomly populating ELGs according to the SHMR and the fraction
$F_{L}\left(M_{\ast}\right)$. The results of the four subsamples $L0$, $L1$,
$L2$ and $L3$ are shown in four panels respectively. The data points with
error bars are measured from VIPERS. The model predictions are plotted as
solid lines. Except for the auto correlations (black), all the other cross
correlations have been multiplied by $2^{n}$ where $n$ changes with color
($n=$ 1 (cyan), 2 (yellow), 3 (lime) and 4 (magenta)) to give a clear
illustration.  Figure 7: Similar to Figure 6, but the lines are the fitting
results of our model with an adjustable satellite fraction $f_{\mathrm{sat}}$.
The best-fitting models as well as their $1\sigma$ error are plotted as solid
lines with shadow areas. The reduced $\chi^{2}$ is also marked in each panel.
Figure 8: Our best-fitting satellite fractions $f_{\mathrm{sat}}$ for the four
$L_{\mathrm{[O\,II]}}$-selected subsamples (red points with error bars). For
comparison, the green circles represent the $f_{\mathrm{sat}}$ if each
$L_{\mathrm{[O\,II]}}$-selected subsample is selected randomly based on the
SHMR and $F_{L}\left(M_{\ast}\right)$.

###  5.2 Populating halos with ELGs

Combining the ELG-stellar mass relation measured above and the SHMR derived in
Section 4, we can connect the ELGs with haloes in the simulation. To model the
auto correlation functions of the $L_{\mathrm{[O\,II]}}$-selected subsamples
and their cross correlation functions with the $M_{\ast}$-selected subsamples,
we also adopt Equation 12 to make the calculation more efficient, just
changing the $P_{x\left(y\right)}\left(M_{\mathrm{h},i}\right)$ and
$P_{x\left(y\right)}\left(M_{\mathrm{s},i}\right)$ in the Equation. For a
$M_{\ast}$-selected subsample $x$, the $P_{x}\left(M_{\mathrm{h},i}\right)$
and $P_{x}\left(M_{\mathrm{s},i}\right)$ can also be calculated by Equation 13
and 14, respectively. And for a $L_{\mathrm{[O\,II]}}$-selected subsample $x$,
we can define its $P_{x}\left(M_{\mathrm{h},i}\right)$ and
$P_{x}\left(M_{\mathrm{s},i}\right)$ as

$\displaystyle\begin{aligned} &P_{x}\left(M_{\mathrm{h},i}\right)\\\
&=P_{\mathrm{cen},x}\left(L^{\mathrm{min}}_{\mathrm{[O\,II]},x}<L_{\mathrm{[O\,II]}}<L^{\mathrm{max}}_{\mathrm{[O\,II]},x}|M_{\mathrm{h},i}\right)\\\
&=\int_{-\infty}^{+\infty}F_{Lx}\left(M_{\ast}\right)p\left(M_{\ast}|M_{\mathrm{h},i}\right)\mathrm{d}M_{\ast}\end{aligned}$
(19)

and

$\displaystyle\begin{aligned} &P_{x}\left(M_{\mathrm{s},i}\right)\\\
&=P_{\mathrm{sat},x}\left(L^{\mathrm{min}}_{\mathrm{[O\,II]},x}<L_{\mathrm{[O\,II]}}<L^{\mathrm{max}}_{\mathrm{[O\,II]},x}|M_{\mathrm{s},i}\right)\\\
&=\int_{-\infty}^{+\infty}F_{Lx}\left(M_{\ast}\right)p\left(M_{\ast}|M_{\mathrm{s},i}\right)\mathrm{d}M_{\ast},\end{aligned}$
(20)

where the $F_{Lx}\left(M_{\ast}\right)$ is the fraction of subsample $x$ at
given $M_{\ast}$ and $p\left(M_{\ast}|M_{\mathrm{h},i}\right)$
($p\left(M_{\ast}|M_{\mathrm{s},i}\right)$) is the SHMR derived in Section 4.
In the above equations, the sample of ELGs is equivalent to a random selection
of the fraction $F_{Lx}\left(M_{\ast}\right)$ of galaxies from the whole
population. In this manner, we calculate the modeled projected cross (auto)
correlation functions $w_{\mathrm{p}}$ for each
$L_{\mathrm{[O\,II]}}$-selected subsample and present them as solid lines in
Figure 6, where the four panels represent the four
$L_{\mathrm{[O\,II]}}$-selected subsamples. We note that our model
overestimates the overall clustering signal, especially at small scales. This
may be caused by the fact that the satellite fraction in each subsample is the
same as that of the normal galaxies. However, in the real Universe, satellite
galaxies are expected form earlier than central ones, so the probability that
they are currently star-forming ELGs is relatively lower. Furthermore, the
modeled satellite fractions $f_{\mathrm{sat}}$ are displayed as green circles
in Figure 8. The value of satellite fraction is close to 0.3, which is
obviously higher than that found in current observational studies (e.g., Guo
et al., 2019; Okumura et al., 2021), in which $f_{\mathrm{sat}}<0.2$.

Motivated by these considerations, we introduce a free parameter
$f_{\mathrm{sat}}$ to modulate the satellite fraction in our model. The
Equation 19 and 20 are re-written as

$\displaystyle\begin{aligned} &P_{x}\left(M_{\mathrm{h},i}\right)\\\
&=P_{\mathrm{cen},x}\left(L^{\mathrm{min}}_{\mathrm{[O\,II]},x}<L_{\mathrm{[O\,II]}}<L^{\mathrm{max}}_{\mathrm{[O\,II]},x}|M_{\mathrm{h},i}\right)\\\
&=\left(1-f_{\mathrm{sat}}\right)\frac{\bar{n}_{\mathrm{h},i}+\bar{n}_{\mathrm{s},i}}{\bar{n}_{\mathrm{h},i}}\times\int_{-\infty}^{+\infty}F_{Lx}\left(M_{\ast}\right)p\left(M_{\ast}|M_{\mathrm{h},i}\right)\mathrm{d}M_{\ast}\end{aligned}$
(21)

and

$\displaystyle\begin{aligned} &P_{x}\left(M_{\mathrm{s},i}\right)\\\
&=P_{\mathrm{sat},x}\left(L^{\mathrm{min}}_{\mathrm{[O\,II]},x}<L_{\mathrm{[O\,II]}}<L^{\mathrm{max}}_{\mathrm{[O\,II]},x}|M_{\mathrm{s},i}\right)\\\
&=f_{\mathrm{sat}}\frac{\bar{n}_{\mathrm{h},i}+\bar{n}_{\mathrm{s},i}}{\bar{n}_{\mathrm{s},i}}\times\int_{-\infty}^{+\infty}F_{Lx}\left(M_{\ast}\right)p\left(M_{\ast}|M_{\mathrm{s},i}\right)\mathrm{d}M_{\ast},\end{aligned}$
(22)

where the $f_{\mathrm{sat}}$ is the satellite fraction. After the SHMR
$p\left(M_{\ast}|M\right)$ and the ELG fraction $F_{L}\left(M_{\ast}\right)$
are fixed, the ELG-halo connection can be determined completely by the
$f_{\mathrm{sat}}$ only.

Next we constrain the parameter $f_{\mathrm{sat}}$ by fitting our model with
the observed cross (auto) correlation functions. Following Equation 16, the
$\chi^{2}$ for the $i$-th $L_{\mathrm{[O\,II]}}$-selected subsample $Li$ can
be written as

$\displaystyle\begin{aligned}
\chi_{Li}^{2}&=\left(w^{LiLi}_{\mathrm{p}}-w^{LiLi}_{\mathrm{p,mod}}\right)^{T}C^{-1}_{LiLi}\left(w^{LiLi}_{\mathrm{p}}-w^{LiLi}_{\mathrm{p,mod}}\right)\\\
&+\sum_{j=0}^{3}\left(w^{LiMj}_{\mathrm{p}}-w^{LiMj}_{\mathrm{p,mod}}\right)^{T}C^{-1}_{LiMj}\left(w^{LiMj}_{\mathrm{p}}-w^{LiMj}_{\mathrm{p,mod}}\right),\end{aligned}$
(23)

where we use one auto correlation function $w^{LiLi}_{\mathrm{p}}$ and four
cross correlation functions $w^{LiMj}_{\mathrm{p}}$ to infer the model
parameter $f_{\mathrm{sat}}$. As a result, the dof in our fitting is
$\mathrm{dof}=12\times 5-1=59$.

We show the best-fitting $f_{\mathrm{sat}}$ as well as the $1\sigma$
dispersion of their posterior distributions in Figure 8. It demonstrates that
the best-fitting $f_{\mathrm{sat}}$ decreases as the $L_{\mathrm{[O\,II]}}$
increases, indicating that the [O II] lines are primarily generated by central
galaxies rather than old satellites with little star formation. The value of
$f_{\mathrm{sat}}$ in our model is also broadly consistent with other
observational results (e.g., Guo et al., 2019; Okumura et al., 2021).

The best-fitting $w_{\mathrm{p}}$ as well as the $1\sigma$ uncertainties are
displayed as the solid lines with shadow regions in Figure 7. The cross
correlation functions of the four ELGs subsamples are well fitted. It suggests
that the SHMR of normal galaxies can be used for ELGs with only the fraction
of satellite galaxies reduced. This may indicate that the clustering of normal
galaxies in the stellar mass range of ELGs does not depend on the star
formation rate. As the ELGs are mostly in the stellar mass range
$<10^{10}\,M_{\odot}$, we expect that normal central galaxies in this mass
range at redshift $z\sim 0.7$ are dominantly star forming galaxies, which
supports why we can use the same SHMR. The lower fraction of the satellites
indicates that the red satellites should not be included in the ELG sample.
Our results are also broadly consistent with the finding of Hadzhiyska et al.
(2021b) that DESI-like ELGs have a small assembly bias based on IllustrisTNG
simulations. In addition, we note that the observed auto correlations of $L2$
and $L3$ are slightly higher than our model predictions especially at small
scales, although the errors are large. In the future, we will carefully
investigate this issue using a much larger ELG sample from DESI.

Figure 9: The cross (auto) correlation functions in redshift-space for the
$L_{\mathrm{[O\,II]}}$-selected subsamples in both the observations and our
models. The four rows from top to bottom represent the four subsamples $L0$,
$L1$, $L2$ and $L3$. The observed monopole $s\xi_{0}\left(s\right)$,
quadrupole $s^{2}\xi_{2}\left(s\right)$ and hexadecapole
$s^{2}\xi_{4}\left(s\right)$ are displayed as data points with error bars in
the three columns from left to right, respectively. Except for the black
points, all the other data points have been shifted by $10\times n$ where $n$
changes with color ($n=$ 1 (cyan), 2 (yellow), 3 (lime) and 4 (magenta)) to
give a clear display. The solid lines represent our model predictions (not
fittings).

###  5.3 Predicting the correlation functions of ELGs in redshift-space

We further check the performance of our model predictions for the clustering
in redshift-space. By replacing the
$w_{\mathrm{p}}\left(r_{\mathrm{p}}\right)$ in Equation 12 with
$\xi_{0}\left(s\right)$, $\xi_{2}\left(s\right)$ and $\xi_{4}\left(s\right)$,
we can calculate these predicted multipole moments, which are presented in
Figure 9 as solid curves. Although we have only fitted the observed real-space
$w_{\mathrm{p}}\left(r_{\mathrm{p}}\right)$, the multipole moments in
redshift-space predicted by our best-fitting model are also in good agreement
with the observations.

In general, our model can well reproduce the cross (auto) correlation
functions in both real-space and redshift-space for the ELGs. This method can
be easily applied to generate ELG mock catalogs for ongoing spectroscopic
surveys such as DESI and PFS.

Figure 10: The comparison of different ELG HOD models. The predicted HODs of
our four [O II] luminosity-selected subsamples are presented as cyan, yellow,
lime and magenta lines. In addition, we present the Geach HOD model (Geach et
al., 2012) with the best-fitting parameters provided by Okumura et al. (2021)
as brown lines. The HOD model proposed by Avila et al. (2020) is also shown as
black lines. The solid, dashed and dotted lines denote the total, central and
satellite occupation numbers, respectively.  Figure 11: Comparison of the
modified Geach HOD form (Equation 25) with the occupations derived from our
mock catalog. The hollow circles denote our derived HODs (same as those shown
in Figure 10) of our mock catalogs for the four $L_{\mathrm{[O\,II]}}$
-selected subsamples. The best-fitting results are shown as solid lines. The
left and right panels correspond to $N_{\mathrm{cen}}$ and $N_{\mathrm{sat}}$
respectively.

###  5.4 Implications for HOD modeling

In order to compare with the traditional HOD models, we work out the HOD based
on our model,

$\displaystyle\begin{aligned}
&N_{\mathrm{cen},x}\left(M_{\mathrm{h}}\right)=P_{x}\left(M_{\mathrm{h}}\right)\\\
&N_{\mathrm{sat},x}\left(M_{\mathrm{h}}\right)=\int
P_{x}\left(M_{\mathrm{s}}\right)n\left(M_{\mathrm{s}}|M_{\mathrm{h}}\right)\mathrm{d}M_{\mathrm{s}}\\\
&N_{x}\left(M_{\mathrm{h}}\right)=N_{\mathrm{cen},x}\left(M_{\mathrm{h}}\right)+N_{\mathrm{sat},x}\left(M_{\mathrm{h}}\right),\end{aligned}$
(24)

where $N_{\mathrm{cen},x}\left(M_{\mathrm{h}}\right)$,
$N_{\mathrm{sat},x}\left(M_{\mathrm{h}}\right)$ and
$N_{x}\left(M_{\mathrm{h}}\right)$ are the central, satellite and total
occupation numbers respectively in the ELG subsample $x$, and the
probabilities $P_{x}\left(M_{\mathrm{h}}\right)$ and
$P_{x}\left(M_{\mathrm{s}}\right)$ are calculated using Equation 21 and 22 but
with a bin width $\Delta\log M=0.1$. Here
$n\left(M_{\mathrm{s}}|M_{\mathrm{h}}\right)$ measured from our simulation is
the mean subhalo mass function at the given $M_{\mathrm{h}}$ bin. The HODs of
the four [O II] luminosity-selected subsamples are shown in Figure 10 as solid
lines, and the decomposed central and satellite occupation numbers are also
displayed as dashed and dotted lines, respectively. Then we compare our model
predictions with two recent HOD models of [O II] ELGs. It should be noted that
since the HOD depends on the target selections of the ELG samples, we can only
qualitatively compare the shapes of these HODs instead of their precise
values.

The first is the Geach HOD model (Geach et al., 2012). Okumura et al. (2021)
has constrained the model parameters based on the HSC NB observations of [O
II] emitters at $z=1.19$ and $z=1.47$, and found that the model can well fit
the angular correlation functions of the [O II] emitters. Considering that the
parameters are better constrained at $=1.47$, we adopt their model parameters
at this redshift based on the posterior PDF (see their Table 3) and display
the HOD as brown lines in Figure 10. The shape of their $N_{\mathrm{cen}}$ at
low-mass end is quite similar to ours. However, with the Geach HOD form,
$N_{\mathrm{cen}}$ in their model tends to be a constant at large halo mass,
while our $N_{\mathrm{cen}}$ keeps decreasing. Although the HOD at massive end
has a relatively small effect on galaxy clustering due to the rapid decline of
the halo mass function, our results imply that a decreasing function
$N_{\mathrm{cen}}=\propto M^{\beta}$ ($\beta\sim-0.2$) can better describe the
massive end of ELG HOD. On the other hand, both the Geach model and ours
present a similar power-law form for $N_{\mathrm{sat}}$.

The other is the HOD model (the HOD-3 in their paper) proposed by Avila et al.
(2020) for the eBOSS ELGs. This model combines a Gaussian function with a
decaying power-law form to describe the central occupation. Avila et al.
(2020) has constrained this model using the semi-analytical model (SAM)
results (Gonzalez-Perez et al., 2018) as well as the eBOSS number density and
bias (see their Table 2). We show this model as black curves in Figure 10.
Although both models show a continuously reduced $N_{\mathrm{cen}}$ towarding
to the massive end, $N_{\mathrm{cen}}$ in the Avila model exhibits a faster
decay after the peak. This difference might imply that the AGN feedback
mechanism in the SAM (Gonzalez-Perez et al., 2018) is too strong, resulting in
quick quenching of galaxies at the massive end. As for $N_{\mathrm{sat}}$, a
power-law form can indeed reasonably describe the $N_{\mathrm{sat}}$ of ELGs
under the current data.

From the above comparison of the three models, we can conclude that since ELGs
are mainly the central galaxies with small stellar mass, the form of
$N_{\mathrm{cen}}$ at large stellar (halo) mass cannot be well constrained
with the clustering data of ELGs only. Our results indicate that the following
form can better describe the HOD of ELGs,

$\displaystyle\begin{aligned}
N_{\mathrm{cen}}\left(M\right)&=N^{\mathrm{Exp}}_{\mathrm{cen,Geach}}\left(M\right)+N^{\mathrm{Erf}}_{\mathrm{cen,Geach}}\left(M\right)\times\left(1+\frac{M}{M_{\mathrm{c}}}\right)^{\beta}\\\
&=F^{\mathrm{B}}_{\mathrm{c}}\left(1-F^{\mathrm{A}}_{\mathrm{c}}\right)\exp\left[-\frac{\log\left(M/M_{\mathrm{c}}\right)^{2}}{2\sigma^{2}_{\log
M}}\right]\\\
&+F^{\mathrm{A}}_{\mathrm{c}}\left[1+\mathrm{erf}\left(\frac{\log\left(M/M_{\mathrm{c}}\right)}{\sigma_{\log
M}}\right)\right]\times\left(1+\frac{M}{M_{\mathrm{c}}}\right)^{\beta}\\\
N_{\mathrm{sat}}\left(M\right)&=N_{\mathrm{sat,Geach}}\left(M\right)\\\
&=F_{\mathrm{s}}\left[1+\mathrm{erf}\left(\frac{\log\left(M/M_{\mathrm{min}}\right)}{\delta_{\log
M}}\right)\right]\left(\frac{M}{M_{\mathrm{min}}}\right)^{\alpha},\end{aligned}$
(25)

where $\beta$ characterizes the decay of $N_{\mathrm{cen}}$ at the high-mass
end. This HOD preserves most features of the Geach HOD form, but remedies it
by introducing a gradually declining function at large mass. On the other
hand, the behavior of $N_{\mathrm{cen}}$ at large stellar (halo) mass can
serve as a sensitive test for feedback models of galaxy formation.

To check the performance of our modified Geach form (Equation 25), we compare
it with our derived HODs of the four $L_{\mathrm{[O\,II]}}$ -selected
subsamples in Figure 11 (the same as those shown in Figure 10, but shown as
hollow circles). The corresponding best-fitting results of the modified HOD
model are displayed as solid lines. We can see that this HOD model can
accurately describe the ELG occupation numbers at all halo mass and at all [O
II] luminosity. The parameters of this HOD model are listed in Table 3.

Table 3: The parameters of the modified Geach form for the four subsamples.  Name | $\log M_{\mathrm{c}}\,[M_{\odot}\,h^{-1}]$ | $\sigma_{\log M}$ | $F^{\mathrm{A}}_{\mathrm{c}}$ | $F^{\mathrm{B}}_{\mathrm{c}}$ | $\beta$ | $\log M_{\mathrm{min}}\,[M_{\odot}\,h^{-1}]$ | $F_{\mathrm{s}}$ | $\delta_{\log M}$ | $\alpha$
---|---|---|---|---|---|---|---|---|---
$L0$ | 11.234 | 0.206 | 0.133 | 0.010 | -0.185 | 11.690 | 0.015 | 0.516 | 0.947
$L1$ | 11.415 | 0.224 | 0.091 | 0.146 | -0.187 | 11.668 | 0.012 | 0.516 | 0.939
$L2$ | 11.528 | 0.241 | 0.035 | 0.075 | -0.168 | 11.723 | 0.005 | 0.508 | 0.940
$L3$ | 11.558 | 0.217 | 0.010 | 0.021 | -0.065 | 11.783 | 0.001 | 0.492 | 0.950

##  6 Summary

In this work, we constrain the ELG-halo connection using the auto and cross
correlation functions of the galaxy subsamples from VIPERS. Combining the SHMR
and ELG-stellar mass distribution, we provide a novel method to populate ELGs
in cosmological simulations. Our main results are summarized as follows.

  1. 1.

Using the galaxy catalog from VIPERS, we construct four stellar mass-selected
subsamples and four [O II] luminosity-selected subsamples. We also take into
account the redshift measurement uncertainty in our N-body simulation to make
a fair comparison with the observations.

  2. 2.

Both the angular and radial selection functions of VIPERS have been carefully
corrected. Particularly, to account for the radial selection effects caused by
the $i$-band limit and the color-color cut, we adopt the $V_{\mathrm{max}}$
technique to generate the redshift distribution for the random samples. For
all the galaxy subsamples, we measure the projected cross (auto) correlation
functions in real-space and the multiple moments in redshift-space.

  3. 3.

To determine the SHMR, we apply the AM model proposed by (Wang & Jing, 2010)
to our N-body simulation. The theoretical cross (auto) correlation functions
of different $M_{\ast}$-selected subsamples are calculated by the tabulated
method. We perform an MCMC analysis to explore the parameters space of SHMR.
Our best-fitting SHMR can recover the observational correlation functions
well.

  4. 4.

We measure the ELG fractions $F_{L}\left(M_{\ast}\right)$ as a function of
stellar mass in the four $L_{\mathrm{[O\,II]}}$-selected subsamples. We
demonstrate that the clustering of ELGs can be well matched both in the real-
space and in the redshift-space if we use the above SHMR to assign stellar
mass to (sub)haloes and then randomly select the ELGs according to their
fractions $F_{L}\left(M_{\ast}\right)$ at a given stellar mass, as long as the
satellite fraction $f_{\mathrm{sat}}$ is properly reduced. The method can be
applied to constructing mock samples for ongoing and future redshift surveys,
such as DESI, PFS and Euclid.

  5. 5.

We also derive the halo occupation numbers for the four ELG subsamples, and
compare them with some of the previous HOD studies for ELGs. Our results
indicate that the Geach form describes well the number of central galaxies at
small and typical halo mass, but its constant form overpredicts the number at
high halo mass. We propose a modified form, Equation 25, for describing HOD of
ELGs. The behavior at the high-mass reflects the feedback processes in galaxy
formation. In addition, the power law form generally describes well the HOD of
satellite galaxies.

In short, the cross correlations between ELGs and normal galaxies can play a
significant role in constraining the ELG-halo connection. For the ongoing and
future spectroscopic surveys, our method can be further developed and tested,
and will provide a novel way to create [O II] ELGs mock catalogs.

H.Y.G thanks Xiaokai Chen for his kind help. The work is supported by NSFC
(12133006, 11890691, 11621303) and by 111 project No. B20019. We gratefully
acknowledge the support of the Key Laboratory for Particle Physics,
Astrophysics and Cosmology, Ministry of Education. This work made use of the
Gravity Supercomputer at the Department of Astronomy, Shanghai Jiao Tong
University. This paper uses data from the VIMOS Public Extragalactic Redshift
Survey (VIPERS). VIPERS has been performed using the ESO Very Large Telescope,
under the ”Large Programme” 182.A-0886. The participating institutions and
funding agencies are listed at http://vipers.inaf.it. Based on observations
collected at the European Southern Observatory, Cerro Paranal, Chile, using
the Very Large Telescope under programs 182.A-0886 and partly 070.A-9007. Also
based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is
operated by the National Research Council (NRC) of Canada, the Institut
National des Sciences de l’Univers of the Centre National de la Recherche
Scientifique (CNRS) of France, and the University of Hawaii. This work is
based in part on data products produced at TERAPIX and the Canadian Astronomy
Data Centre as part of the CanadaFrance-Hawaii Telescope Legacy Survey, a
collaborative project of NRC and CNRS. This research uses data from the VIMOS
VLT Deep Survey, obtained from the VVDS database operated by Cesam,
Laboratoire d’Astrophysique de Marseille, France.

Figure 12: The angular completeness function of VIPERS. In the top panel, the
data points with error bar show the angular correlation functions of parent
$w_{\mathrm{p}}\left(\theta\right)$ (green) and spectroscopic
$w_{\mathrm{s}}\left(\theta\right)$ (blue) samples measured from 153 VIPERS
mock catalog. The angular completeness function defined as
$C\left(\theta\right)=\left[1+w_{\mathrm{s}}\left(\theta\right)\right]/\left[1+w_{\mathrm{p}}\left(\theta\right)\right]$
is plotted as orange points in the bottom panel. The orange solid line
represents the linear interpolation of $C\left(\theta\right)-\log\theta$
relation.

##  Appendix A Correction to the small-scale clustering

Using the 153 VIPERS mock samples provided by Pezzotta et al. (2017), we
compute the angular correlation function $w_{\mathrm{p}}\left(\theta\right)$
for parent photometric galaxies and $w_{\mathrm{s}}\left(\theta\right)$ for
spectroscopic galaxies which is obtained by applying the silt assign algorithm
to the parent catalog. In Figure 12, we show the measurements of
$w_{\mathrm{p}}\left(\theta\right)$ ($w_{\mathrm{s}}\left(\theta\right)$) in
the top panel and
$C\left(\theta\right)=\left[1+w_{\mathrm{s}}\left(\theta\right)\right]/\left[1+w_{\mathrm{p}}\left(\theta\right)\right]$
in the bottom panel. The two turning points of $C\left(\theta\right)$ clearly
reflect the two typical scales that affect the small-scale clustering. We
linearly interpolate the $C\left(\theta\right)-\log\theta$ relation and define
the angular weight as
$w^{\mathrm{A}}\left(\theta\right)=1/C\left(\theta\right)$.

Figure 13: The color space distributions of the random points generated for
the full galaxy sample. The blue (red) points denote the random points with
$z<0.5$ ($z>0.5$). The lime solid line represents the VIPERS color-color cut
(Equation 1) used to exclude the galaxies with $z<0.5$.  Figure 14: The radial
distributions of the $M_{\ast}$-selected subsamples. Similar to Figure 2, the
un-weighted (weighted) number densities as function of comoving distance for
different subsamples are shown as blue (green) histograms. The radial
distributions of corresponding random subsamples are plotted as red solid
lines.  Figure 15: Similar to Figure 14, the radial distributions of the
$L_{\mathrm{[O\,II]}}$ -selected subsamples.

##  Appendix B Generating the redshift distribution for random sample

The $V_{\mathrm{max}}$ method (Cole, 2011; de la Torre et al., 2013; Pezzotta
et al., 2017; de la Torre et al., 2017; Yang et al., 2020) is adopted to
produce the radial distribution for random samples. It is in principle much
better than the method of randomly shuffling the observed redshifts in
generating a random sample. For each galaxy, we convert its $z_{\mathrm{max}}$
output by LE PHARE (Arnouts et al., 2002; Ilbert et al., 2006) into
$V_{\mathrm{max}}$, which represents the maximum volume of this galaxy that
can be observed in VIPERS. Consider that the SSR and TSR have slight impact on
the redshift distribution of sample (Pezzotta et al., 2017), we weight each
galaxy by $w^{\mathrm{SSR}}\times w^{\mathrm{TSR}}$. Then we randomly select
$N_{\mathrm{ran}}$ galaxies based on their probabilities (the probability of
selecting different galaxies is not equal due to the weight). For each
selected galaxy, we can generate a random point uniformly distributed in its
$V_{\mathrm{max}}$ and convert the volume $V_{\mathrm{ran}}$ of this random
point into its redshift $z_{\mathrm{ran}}$. As a result, the original best-
fitting SED of the galaxy should be shifted to $z_{\mathrm{ran}}$ as a new SED
of this random point. We can calculate the $u,g,r,i,z$ magnitudes for this
random point and apply the color-color cut (Equation 1) to it. The Figure 13
presents the color space distributions of the random points generated by the
above method for the full galaxy sample. We note that the color cut (lime
solid line) can clearly distinguish random samples with redshifts lower than
(blue points) and higher than (red points) 0.5. The radial distributions of
random samples generated by this non-parametric method are shown in Figure 2
for the total sample and in Figure 14, and 15 for the subsamples.

## References

  * Arnouts et al. (2002) Arnouts, S., Moscardini, L., Vanzella, E., et al. 2002, MNRAS, 329, 355, doi: 10.1046/j.1365-8711.2002.04988.x
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Avila et al. (2020) Avila, S., Gonzalez-Perez, V., Mohammad, F. G., et al. 2020, MNRAS, 499, 5486, doi: 10.1093/mnras/staa2951
  * Behroozi et al. (2019) Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182
  * Behroozi et al. (2010) Behroozi, P. S., Conroy, C., & Wechsler, R. H. 2010, ApJ, 717, 379, doi: 10.1088/0004-637X/717/1/379
  * Berlind & Weinberg (2002) Berlind, A. A., & Weinberg, D. H. 2002, ApJ, 575, 587, doi: 10.1086/341469
  * Bryan & Norman (1998) Bryan, G. L., & Norman, M. L. 1998, ApJ, 495, 80, doi: 10.1086/305262
  * Chaves-Montero et al. (2016) Chaves-Montero, J., Angulo, R. E., Schaye, J., et al. 2016, MNRAS, 460, 3100, doi: 10.1093/mnras/stw1225
  * Cole (2011) Cole, S. 2011, MNRAS, 416, 739, doi: 10.1111/j.1365-2966.2011.19093.x
  * Cole et al. (2005) Cole, S., Percival, W. J., Peacock, J. A., et al. 2005, MNRAS, 362, 505, doi: 10.1111/j.1365-2966.2005.09318.x
  * Comparat et al. (2013) Comparat, J., Kneib, J.-P., Escoffier, S., et al. 2013, MNRAS, 428, 1498, doi: 10.1093/mnras/sts127
  * Comparat et al. (2016) Comparat, J., Zhu, G., Gonzalez-Perez, V., et al. 2016, MNRAS, 461, 1076, doi: 10.1093/mnras/stw1393
  * Contreras et al. (2013) Contreras, S., Baugh, C. M., Norberg, P., & Padilla, N. 2013, MNRAS, 432, 2717, doi: 10.1093/mnras/stt629
  * Cooper et al. (2010) Cooper, M. C., Gallazzi, A., Newman, J. A., & Yan, R. 2010, MNRAS, 402, 1942, doi: 10.1111/j.1365-2966.2009.16020.x
  * Cooray (2006) Cooray, A. 2006, MNRAS, 365, 842, doi: 10.1111/j.1365-2966.2005.09747.x
  * Cui et al. (2021) Cui, W., Davé, R., Peacock, J. A., Anglés-Alcázar, D., & Yang, X. 2021, Nature Astronomy, 5, 1069, doi: 10.1038/s41550-021-01404-1
  * Davidzon et al. (2013) Davidzon, I., Bolzonella, M., Coupon, J., et al. 2013, A&A, 558, A23, doi: 10.1051/0004-6361/201321511
  * Davis et al. (1985) Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371, doi: 10.1086/163168
  * Davis & Peebles (1983) Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465, doi: 10.1086/160884
  * Dawson et al. (2016) Dawson, K. S., Kneib, J.-P., Percival, W. J., et al. 2016, AJ, 151, 44, doi: 10.3847/0004-6256/151/2/44
  * de la Torre et al. (2013) de la Torre, S., Guzzo, L., Peacock, J. A., et al. 2013, A&A, 557, A54, doi: 10.1051/0004-6361/201321463
  * de la Torre et al. (2017) de la Torre, S., Jullo, E., Giocoli, C., et al. 2017, A&A, 608, A44, doi: 10.1051/0004-6361/201630276
  * DESI Collaboration et al. (2016) DESI Collaboration, Aghamousa, A., Aguilar, J., et al. 2016, arXiv e-prints, arXiv:1611.00036.  https://arxiv.org/abs/1611.00036
  * Eisenstein et al. (2005) Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, ApJ, 633, 560, doi: 10.1086/466512
  * Favole et al. (2017) Favole, G., Rodríguez-Torres, S. A., Comparat, J., et al. 2017, MNRAS, 472, 550, doi: 10.1093/mnras/stx1980
  * Favole et al. (2016) Favole, G., Comparat, J., Prada, F., et al. 2016, MNRAS, 461, 3421, doi: 10.1093/mnras/stw1483
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Gao & Jing (2021) Gao, H., & Jing, Y. P. 2021, ApJ, 908, 43, doi: 10.3847/1538-4357/abd40f
  * Garilli et al. (2014) Garilli, B., Guzzo, L., Scodeggio, M., et al. 2014, A&A, 562, A23, doi: 10.1051/0004-6361/201322790
  * Geach et al. (2012) Geach, J. E., Sobral, D., Hickox, R. C., et al. 2012, MNRAS, 426, 679, doi: 10.1111/j.1365-2966.2012.21725.x
  * Gonzalez-Perez et al. (2018) Gonzalez-Perez, V., Comparat, J., Norberg, P., et al. 2018, MNRAS, 474, 4024, doi: 10.1093/mnras/stx2807
  * Grogin et al. (2011) Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35, doi: 10.1088/0067-0049/197/2/35
  * Gunn & Gott (1972) Gunn, J. E., & Gott, J. Richard, I. 1972, ApJ, 176, 1, doi: 10.1086/151605
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332, doi: 10.1086/500975
  * Guo et al. (2018) Guo, H., Yang, X., & Lu, Y. 2018, ApJ, 858, 30, doi: 10.3847/1538-4357/aabc56
  * Guo et al. (2016) Guo, H., Zheng, Z., Behroozi, P. S., et al. 2016, MNRAS, 459, 3040, doi: 10.1093/mnras/stw845
  * Guo et al. (2019) Guo, H., Yang, X., Raichoor, A., et al. 2019, ApJ, 871, 147, doi: 10.3847/1538-4357/aaf9ad
  * Guo & White (2014) Guo, Q., & White, S. 2014, MNRAS, 437, 3228, doi: 10.1093/mnras/stt2116
  * Guo et al. (2010) Guo, Q., White, S., Li, C., & Boylan-Kolchin, M. 2010, MNRAS, 404, 1111, doi: 10.1111/j.1365-2966.2010.16341.x
  * Guzzo et al. (2014) Guzzo, L., Scodeggio, M., Garilli, B., et al. 2014, A&A, 566, A108, doi: 10.1051/0004-6361/201321489
  * Hadzhiyska et al. (2021a) Hadzhiyska, B., Bose, S., Eisenstein, D., & Hernquist, L. 2021a, MNRAS, 501, 1603, doi: 10.1093/mnras/staa3776
  * Hadzhiyska et al. (2020) Hadzhiyska, B., Bose, S., Eisenstein, D., Hernquist, L., & Spergel, D. N. 2020, MNRAS, 493, 5506, doi: 10.1093/mnras/staa623
  * Hadzhiyska et al. (2021b) Hadzhiyska, B., Tacchella, S., Bose, S., & Eisenstein, D. J. 2021b, MNRAS, 502, 3599, doi: 10.1093/mnras/stab243
  * Hamilton (1992) Hamilton, A. J. S. 1992, ApJ, 385, L5, doi: 10.1086/186264
  * Han et al. (2018) Han, J., Cole, S., Frenk, C. S., Benitez-Llambay, A., & Helly, J. 2018, MNRAS, 474, 604, doi: 10.1093/mnras/stx2792
  * Han et al. (2012) Han, J., Jing, Y. P., Wang, H., & Wang, W. 2012, MNRAS, 427, 2437, doi: 10.1111/j.1365-2966.2012.22111.x
  * Hartlap et al. (2007) Hartlap, J., Simon, P., & Schneider, P. 2007, A&A, 464, 399, doi: 10.1051/0004-6361:20066170
  * Hearin et al. (2015) Hearin, A. P., Watson, D. F., & van den Bosch, F. C. 2015, MNRAS, 452, 1958, doi: 10.1093/mnras/stv1358
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science Engineering, 9, 90
  * Ilbert et al. (2006) Ilbert, O., Arnouts, S., McCracken, H. J., et al. 2006, A&A, 457, 841, doi: 10.1051/0004-6361:20065138
  * Jarvis et al. (2013) Jarvis, M. J., Bonfield, D. G., Bruce, V. A., et al. 2013, MNRAS, 428, 1281, doi: 10.1093/mnras/sts118
  * Jiang et al. (2008) Jiang, C. Y., Jing, Y. P., Faltenbacher, A., Lin, W. P., & Li, C. 2008, ApJ, 675, 1095, doi: 10.1086/526412
  * Jing (2019) Jing, Y. 2019, Science China Physics, Mechanics, and Astronomy, 62, 19511, doi: 10.1007/s11433-018-9286-x
  * Jing et al. (1998) Jing, Y. P., Mo, H. J., & Börner, G. 1998, ApJ, 494, 1, doi: 10.1086/305209
  * Jing & Suto (2002) Jing, Y. P., & Suto, Y. 2002, ApJ, 574, 538, doi: 10.1086/341065
  * Kaiser (1987) Kaiser, N. 1987, MNRAS, 227, 1, doi: 10.1093/mnras/227.1.1
  * Kocevski et al. (2011) Kocevski, D. D., Lemaux, B. C., Lubin, L. M., et al. 2011, ApJ, 737, L38, doi: 10.1088/2041-8205/737/2/L38
  * Koekemoer et al. (2011) Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApJS, 197, 36, doi: 10.1088/0067-0049/197/2/36
  * Landy & Szalay (1993) Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64, doi: 10.1086/172900
  * Le Fèvre et al. (2003) Le Fèvre, O., Saisse, M., Mancini, D., et al. 2003, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Commissioning and performances of the VLT-VIMOS instrument, ed. M. Iye & A. F. M. Moorwood, 1670–1681, doi: 10.1117/12.460959
  * Levenberg (1944) Levenberg, K. 1944, Quarterly of applied mathematics, 2, 164
  * Ma & Fry (2000) Ma, C.-P., & Fry, J. N. 2000, ApJ, 543, 503, doi: 10.1086/317146
  * Mandelbaum et al. (2016) Mandelbaum, R., Wang, W., Zu, Y., et al. 2016, MNRAS, 457, 3200, doi: 10.1093/mnras/stw188
  * Marquardt (1963) Marquardt, D. W. 1963, Journal of the society for Industrial and Applied Mathematics, 11, 431
  * Martin et al. (2005) Martin, D. C., Fanson, J., Schiminovich, D., et al. 2005, ApJ, 619, L1, doi: 10.1086/426387
  * McLeod et al. (2021) McLeod, D. J., McLure, R. J., Dunlop, J. S., et al. 2021, MNRAS, 503, 4413, doi: 10.1093/mnras/stab731
  * Moster et al. (2013) Moster, B. P., Naab, T., & White, S. D. M. 2013, MNRAS, 428, 3121, doi: 10.1093/mnras/sts261
  * Moutard et al. (2016) Moutard, T., Arnouts, S., Ilbert, O., et al. 2016, A&A, 590, A102, doi: 10.1051/0004-6361/201527945
  * Newman et al. (2013) Newman, J. A., Cooper, M. C., Davis, M., et al. 2013, ApJS, 208, 5, doi: 10.1088/0067-0049/208/1/5
  * Okumura et al. (2021) Okumura, T., Hayashi, M., Chiu, I. N., et al. 2021, PASJ, 73, 1186, doi: 10.1093/pasj/psab068
  * Oliphant (2007) Oliphant, T. E. 2007, Computing in Science Engineering, 9, 10
  * Osato et al. (2021) Osato, K., Nishimichi, T., & Takada, M. 2021, arXiv e-prints, arXiv:2107.13168.  https://arxiv.org/abs/2107.13168
  * Peacock & Smith (2000) Peacock, J. A., & Smith, R. E. 2000, MNRAS, 318, 1144, doi: 10.1046/j.1365-8711.2000.03779.x
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
  * Pezzotta et al. (2017) Pezzotta, A., de la Torre, S., Bel, J., et al. 2017, A&A, 604, A33, doi: 10.1051/0004-6361/201630295
  * Pozzetti et al. (2007) Pozzetti, L., Bolzonella, M., Lamareille, F., et al. 2007, A&A, 474, 443, doi: 10.1051/0004-6361:20077609
  * Pozzetti et al. (2010) Pozzetti, L., Bolzonella, M., Zucca, E., et al. 2010, A&A, 523, A13, doi: 10.1051/0004-6361/200913020
  * Raichoor et al. (2017) Raichoor, A., Comparat, J., Delubac, T., et al. 2017, MNRAS, 471, 3955, doi: 10.1093/mnras/stx1790
  * Rodríguez-Puebla et al. (2015) Rodríguez-Puebla, A., Avila-Reese, V., Yang, X., et al. 2015, ApJ, 799, 130, doi: 10.1088/0004-637X/799/2/130
  * Rodríguez-Torres et al. (2016) Rodríguez-Torres, S. A., Chuang, C.-H., Prada, F., et al. 2016, MNRAS, 460, 1173, doi: 10.1093/mnras/stw1014
  * Schechter (1976) Schechter, P. 1976, ApJ, 203, 297, doi: 10.1086/154079
  * Scodeggio et al. (2018) Scodeggio, M., Guzzo, L., Garilli, B., et al. 2018, A&A, 609, A84, doi: 10.1051/0004-6361/201630114
  * Seljak (2000) Seljak, U. 2000, MNRAS, 318, 203, doi: 10.1046/j.1365-8711.2000.03715.x
  * Simha et al. (2012) Simha, V., Weinberg, D. H., Davé, R., et al. 2012, MNRAS, 423, 3458, doi: 10.1111/j.1365-2966.2012.21142.x
  * Szapudi & Szalay (1998) Szapudi, I., & Szalay, A. S. 1998, ApJ, 494, L41, doi: 10.1086/311146
  * Takada et al. (2014) Takada, M., Ellis, R. S., Chiba, M., et al. 2014, PASJ, 66, R1, doi: 10.1093/pasj/pst019
  * Vale & Ostriker (2006) Vale, A., & Ostriker, J. P. 2006, MNRAS, 371, 1173, doi: 10.1111/j.1365-2966.2006.10605.x
  * van den Bosch et al. (2007) van den Bosch, F. C., Yang, X., Mo, H. J., et al. 2007, MNRAS, 376, 841, doi: 10.1111/j.1365-2966.2007.11493.x
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science Engineering, 13, 22
  * Wang & Jing (2010) Wang, L., & Jing, Y. P. 2010, MNRAS, 402, 1796, doi: 10.1111/j.1365-2966.2009.16007.x
  * Wang et al. (2006) Wang, L., Li, C., Kauffmann, G., & De Lucia, G. 2006, MNRAS, 371, 537, doi: 10.1111/j.1365-2966.2006.10669.x
  * Wang et al. (2013) Wang, L., Weinmann, S. M., De Lucia, G., & Yang, X. 2013, MNRAS, 433, 515, doi: 10.1093/mnras/stt743
  * Wang et al. (2019) Wang, Z., Xu, H., Yang, X., et al. 2019, ApJ, 879, 71, doi: 10.3847/1538-4357/ab25e6
  * Wechsler et al. (1998) Wechsler, R. H., Gross, M. A. K., Primack, J. R., Blumenthal, G. R., & Dekel, A. 1998, ApJ, 506, 19, doi: 10.1086/306229
  * Wechsler & Tinker (2018) Wechsler, R. H., & Tinker, J. L. 2018, ARA&A, 56, 435, doi: 10.1146/annurev-astro-081817-051756
  * Xu et al. (2016) Xu, H., Zheng, Z., Guo, H., Zhu, J., & Zehavi, I. 2016, MNRAS, 460, 3647, doi: 10.1093/mnras/stw1259
  * Xu et al. (2018) Xu, H., Zheng, Z., Guo, H., et al. 2018, MNRAS, 481, 5470, doi: 10.1093/mnras/sty2615
  * Xu & Jing (2021) Xu, K., & Jing, Y. 2021, arXiv e-prints, arXiv:2110.05760.  https://arxiv.org/abs/2110.05760
  * Xu et al. (2021) Xu, K., Zheng, Y., & Jing, Y. 2021, arXiv e-prints, arXiv:2109.11738.  https://arxiv.org/abs/2109.11738
  * Xu & Zheng (2020) Xu, X., & Zheng, Z. 2020, MNRAS, 492, 2739, doi: 10.1093/mnras/staa009
  * Yang et al. (2020) Yang, L., Jing, Y.-P., Li, Z.-G., & Yang, X.-H. 2020, Research in Astronomy and Astrophysics, 20, 054, doi: 10.1088/1674-4527/20/4/54
  * Yang et al. (2003) Yang, X., Mo, H. J., & van den Bosch, F. C. 2003, MNRAS, 339, 1057, doi: 10.1046/j.1365-8711.2003.06254.x
  * Yang et al. (2009) —. 2009, ApJ, 695, 900, doi: 10.1088/0004-637X/695/2/900
  * Yang et al. (2012) Yang, X., Mo, H. J., van den Bosch, F. C., Zhang, Y., & Han, J. 2012, ApJ, 752, 41, doi: 10.1088/0004-637X/752/1/41
  * York et al. (2000) York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513
  * Yuan et al. (2018) Yuan, S., Eisenstein, D. J., & Garrison, L. H. 2018, MNRAS, 478, 2019, doi: 10.1093/mnras/sty1089
  * Zehavi et al. (2011) Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59, doi: 10.1088/0004-637X/736/1/59
  * Zentner et al. (2014) Zentner, A. R., Hearin, A. P., & van den Bosch, F. C. 2014, MNRAS, 443, 3044, doi: 10.1093/mnras/stu1383
  * Zheng et al. (2007) Zheng, Z., Coil, A. L., & Zehavi, I. 2007, ApJ, 667, 760, doi: 10.1086/521074
  * Zheng & Guo (2016) Zheng, Z., & Guo, H. 2016, MNRAS, 458, 4015, doi: 10.1093/mnras/stw523
  * Zheng et al. (2005) Zheng, Z., Berlind, A. A., Weinberg, D. H., et al. 2005, ApJ, 633, 791, doi: 10.1086/466510
  * Zu & Mandelbaum (2015) Zu, Y., & Mandelbaum, R. 2015, MNRAS, 454, 1161, doi: 10.1093/mnras/stv2062
  * Zu & Mandelbaum (2016) —. 2016, MNRAS, 457, 4360, doi: 10.1093/mnras/stw221
  * Zu & Mandelbaum (2018) —. 2018, MNRAS, 476, 1637, doi: 10.1093/mnras/sty279
  * Zu et al. (2021a) Zu, Y., Shan, H., Zhang, J., et al. 2021a, MNRAS, 505, 5117, doi: 10.1093/mnras/stab1712
  * Zu et al. (2021b) Zu, Y., Song, Y., Shao, Z., et al. 2021b, arXiv e-prints, arXiv:2108.06790.  https://arxiv.org/abs/2108.06790