The DESI One-Percent survey: constructing galaxy-halo connections for ELGs and LRGs using auto and cross correlations

In the current Dark Energy Spectroscopic Instrument (DESI) survey, emission line galaxies (ELGs) and luminous red galaxies (LRGs) are essential for mapping the dark matter distribution at z ∼ 1. We measure the auto and cross correlation functions of ELGs and LRGs at 0 . 8 < z ≤ 1 . 0 from the DESI One-Percent survey. Following Gao et al. (2022), we construct the galaxy-halo connections for ELGs and LRGs simultaneously. With the stellar-halo mass relation (SHMR) for the whole galaxy population (i.e. normal galaxies), LRGs can be selected directly by stellar mass, while ELGs can also be selected randomly based on the observed number density of each stellar mass, once the probability P sat of a satellite galaxy becoming an ELG is determined. We demonstrate that the observed small scale clustering prefers a halo mass-dependent P sat model rather than a constant. With this model, we can well reproduce the auto correlations of LRGs and the cross correlations between LRGs and ELGs at r p > 0 . 1 Mpc h − 1 . We can also reproduce the auto correlations of ELGs at r p > 0 . 3 Mpc h − 1 ( s > 1 Mpc h − 1 ) in real (redshift) space. Although our model has only seven parameters, we show that it can be extended to higher redshifts and reproduces the observed auto correlations of ELGs in the whole range of 0 . 8 < z < 1 . 6, which enables us to generate a lightcone ELG mock for DESI. With the above model, we further derive halo occupation distributions (HODs) for ELGs which can be used to produce ELG mocks in coarse simulations without resolving subhalos.


INTRODUCTION
A precise understanding of the connection between galaxies and dark matter is one of the most critical challenges in current research.Galaxies are formed in dark matter halos, and the growth of galaxies is closely related to the growth of their host halos (Wechsler & Tinker 2018).Thus, establishing the galaxy-halo connection is a prerequisite for understanding galaxy formation and evolution.Moreover, cosmological probes, such as baryon acoustic oscillation (BAO, e.g., Cole et al. 2005;Eisenstein et al. 2005), redshift-space distortion (RSD, e.g., Kaiser 1987), and weak gravitational lensing (e.g., Bartelmann & Schneider 2001;Mandelbaum 2018) provide us with powerful ways to infer the cosmological parameters and constrain the dark energy model.To make these cosmological probes accurate enough to fulfill the requirements of current cosmological studies, an accurate relation between the galaxies and the underlying dark matter halos is needed.
Galaxy samples complete to a stellar mass (or a broadband luminosity) are typically required in previous studies of the galaxy-halo connection.These samples are usually constructed, with proper incompleteness corrections, from flux-limited redshift surveys such as the Sloan Digital Sky Survey (SDSS, York et al. 2000;Gunn et al. 2006), the VIMOS VLT Deep Survey (VVDS, York et al. 2000;Gunn et al. 2006), the Deep Extragalactic Evolutionary Probe 2 (DEEP2, Newman et al. 2013) and the VIMOS Public Extragalactic Redshift Survey (VIPERS, Guzzo et al. 2014;Garilli et al. 2014;Scodeggio et al. 2018).Since these galaxies represent the general population, we refer to them as normal galaxies in this paper.Above a certain stellar mass threshold, these normal galaxies are complete, and thus their clustering and SMF can be used to constrain the SHMR.However, at medium and high redshifts, due to the limited wavelength coverage and detection depth, galaxies are usually color and magnitude selected, such as the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al. 2013) and Dark Energy Spectroscopic Instrument (DESI, Levi et al. 2013;DESI Collaboration et al. 2016a,b, 2022) surveys, which makes it challenging to construct the stellar mass (or luminosity) limited samples.Recently, Photometric objects Around Cosmic webs (PAC, Xu et al. 2022b) method has been proposed to overcome this difficulty.Utilizing the correlations between the photometric data from the DESI Legacy Imaging Survey (Dey et al. 2019) and spectroscopic samples at various redshifts, Xu et al. (2023) greatly improved the SHMR measurements down to 10 8.0 M at z ∼ 0.2 and 10 9.8 M at z ∼ 0.7.Nevertheless, at z ∼ 1, PAC still requires a large sample of galaxies with deeper photometric observations over the wide sky area of a redshift survey.
As galaxy surveys are extended to higher redshifts, galaxies with specific spectral features have become the main targets of current spectroscopic surveys, such as SDSS IV extended Baryon Oscillation Spectroscopic Survey (eBOSS, Dawson et al. 2016), DESI (Levi et al. 2013;DESI Collaboration et al. 2016a,b, 2022) and Subaru Prime Focus Spectrograph (PFS, Takada et al. 2014;Tamura et al. 2022).In particular, DESI achieves coverage of the sky area over 14,000 deg 2 , and is devoted to targeting more than 8 million luminous red galaxies (LRGs) at 0.4 < z < 1.0 and 16 million [O II] emission line galaxies (ELGs) at 0.6 < z < 1.6.The combination of these two types of galaxy samples will provide us with an invaluable opportunity to study the galaxy-halo connection at z ∼ 1.However, it is a big challenge to accurately model the galaxy-halo connection for these targeted galaxies and constrain the overall SHMR, since they are incomplete for stellar mass limits.The incompleteness could be very complicated due to their complex color and magnitude selection especially for ELGs (see Raichoor et al. (2023)).Different from the normal galaxy population, many studies (e.g., Geach et al. 2012;Contreras et al. 2013;Favole et al. 2016Favole et al. , 2017;;Gonzalez-Perez et al. 2018;Guo et al. 2019;Gonzalez-Perez et al. 2020;Avila et al. 2020;Hadzhiyska et al. 2021;Zhai et al. 2021;Okumura et al. 2021;Yuan et al. 2022b;Hadzhiyska et al. 2022a,b;Lin et al. 2023) have proved that the HOD of ELGs is expected to peak at some host halo mass (∼ 10 12 M ) and decrease as the host halo mass increases, but the HOD forms obtained vary widely.
More and more studies focus on improving the galaxyhalo connection models for LRG and ELG samples at z > 0.5.For example, by jointly modeling the BOSS LRGs and eBOSS ELGs, Guo et al. (2019) adopted the incomplete conditional stellar mass function (IC-SMF) model to constrain the completeness of ELG, the galaxy quenched fraction and the SHMR down to ∼ 10 10 M .For efficient HOD analysis, Yuan et al. (2022b) developed a multi-tracer HOD framework that can model the cross correlations of LRGs, ELGs and Quasars (QSOs) as well as their environment-based secondary galaxy bias.But the combination of different HOD models for the three galaxy tracers introduces a large number of parameters, which increases the difficulty of the precise constraints of the parameters.Using the DESI-like mock samples from a hydrodynamical simulation, Hadzhiyska et al. (2022a,b) optimized the HODs for LRG and ELG through modeling galaxy conformity effects and improved the theoretical predictions of clustering on both small and large scales, though the current generation of hydrodynamical simulations still lack the power to accurately predict the properties of galaxies as required by the current precise cosmological studies.As we will show later, for the complicated target selection, the HOD of ELGs in DESI has a so complicated dependence on redshift that it is extremely challenging to propose an analytical expression, in contrast to that for normal galaxies.
Instead of using the conventional HOD approach, Gao et al. (2022) (hereafter Paper I) has proposed a novel SHAM approach to construct the SHMR and the galaxyhalo connection for ELGs.They measured the auto and cross correlations between ELGs and the stellar massselected normal galaxy samples from VIPERS.They determined the SHMR for normal galaxies using the abundance and clustering of stellar mass-selected samples.They then proposed that ELGs could be randomly selected from the normal galaxy population, as long as the ELG satellite fraction is reasonably reduced, and the satellite fraction changes with the strength of the [O II] emission line.They demonstrate that this approach can well reproduce the auto and cross correlations for ELGs in both real-space and redshift-space.The main advantage of this approach is that the fundamental relation between the galaxy and the host halo (or subhalo) is determined by stellar mass, and only the satellite fraction is a free parameter for ELGs.Other studies (e.g., Rodríguez-Puebla et al. 2015;Wang et al. 2021;Zhang et al. 2022) also implied that SHMR for ELGs is similar to that for normal galaxies, since the star-forming Gao et al. galaxies dominate the whole population at the relevant stellar mass range (M * < 10 10.5 M ) and the host halo mass only weakly depends on galaxy color when stellar mass is fixed (see in particular Figure 10 of Rodríguez-Puebla et al. (2015)).
The LRG and ELG samples with high spectroscopic completeness from the One-Percent survey of the DESI Survey Validation (SV, DESI Collaboration et al. 2023a) enable us to further refine the SHAM method in Paper I and to extend the galaxy-halo connection model to redshift z ∼ 1.On the one hand, by combining the LRG and ELG samples, we can probe the SHMR at both the low and high mass ends.LRGs dominate the high mass end (M * > 10 11 M ) of the SMF, and ELGs are star-forming blue galaxies with low and intermediate mass (M * < 10 10.5 M ).These two types of galaxies cover a wide range of stellar masses.On the other hand, although ELGs have a wide range of host halo masses, most central ELGs are expected to be located in small halos with M h < 10 12.5 M .The LRGxELG cross-correlation can help to reveal the distribution of ELGs around the massive halos, that is, the distribution of ELGs as satellites in massive halos.Therefore, with the cross-correlations of the overlapping LRG and ELG samples, in addition to their auto correlations, we can achieve a stronger constraint on the ELG-halo connection.
In this work, we will first measure the auto and cross correlations of the LRG and ELG samples from the One-Percent survey, as well as their observed number densities.Then, following Paper I, we will simultaneously determine both the SHMR for normal galaxies and the ELG-halo connection.We will demonstrate that after modeling the normal galaxies in simulation using the SHMR, LRGs can be selected from the massive normal galaxies, while ELGs can be selected randomly from the normal galaxies based on the observed number density after reducing the fraction of the satellite galaxies, which is a function of the host halo mass.With our models, we will develop a method to generate an ELG mock sample that has the same number density and clustering properties as the DESI ELG sample.We expect that the mock samples will be very useful for future cosmological studies based on the DESI ELGs.Finally, it is straightforward to generate mock samples for the DESI LRG sample, since the target selection criterion is relatively simple.
The paper is structured as follows.In Section 2, we describe the observed galaxy samples and show our measurements of galaxy clustering.In Section 3, we present our basic ideas for modeling the galaxy-halo connection using the N-body simulation.The fitting results are pre-  sented in Section 4. In Section 5, we derive the HOD for ELG based on our ELG-halo connection.Finally, we briefly summarize the main results of this work.The cosmological parameters used in the calculations and simulations in this paper are Ω m,0 = 0.268, Ω Λ,0 = 0.732 and Our work is one of many studies for the galaxy-halo connections in the One-Percent survey.Other relevant parallel studies include: HOD modeling for ELGs (Rocher et al. 2023;Lasker et al. 2023), for LRGs (Ereza et al. 2023;Yuan et al. 2023), for QSOs (Rajeev et al. 2023;Yuan et al. 2023), BGS HOD (Grove et al. 2023;Smith et al. 2023), and for multi-tracers (Yuan et al. 2023), and SHAM modeling for the different tracers (Yu et al. 2023;Prada et al. 2023).The combined efforts will greatly improve the current understanding of the galaxy-halo connection for the different tracers in DESI.] ELG 0.6 < z 0.7 0.7 < z 0.8 0.8 < z 0.9 0.9 < z 1.0  In this section, we briefly introduce the One-Percent survey.We describe the selection of LRG and ELG subsamples, and show their observed number densities as a function of stellar mass and redshift.The measurements of the auto and cross correlation functions are also presented.

DESI One-Percent survey
DESI is dedicated to collecting the spectra for approximately 40 million extra-galactic objects covering more than 14,000 deg 2 in five years (Levi et al. 2013;DESI Collaboration et al. 2016a,b, 2022).The spectroscopic observations of DESI are performed by a multi-object, fiber-fed spectrograph attached to the prime focus panel of the 4-meter Mayall telescope at Kitt Peak National Observatory (DESI Collaboration et al. 2022).The spectrograph spans a wavelength range of 3600 − 9800 Å and can assign fibers to 5,000 objects at a time (DESI Collaboration et al. 2016b;Silber et al. 2023;Miller et al. 2023).The multiple supporting pipelines of DESI experiment are described in detail by Guy et al. (2023); Bailey et al. (2023); Raichoor et al. (2023); Schlafly et al. (2023); Myers et al. (2023).The target selections and survey validations of DESI can be found in a series of papers (Allende Prieto et al. 2020;Ruiz-Macias et al. 2020;Zhou et al. 2020;Raichoor et al. 2020;Yèche et al. 2020;Lan et al. 2023;Alexander et al. 2023;Cooper et al. 2023;Hahn et al. 2023;Chaussidon et al. 2023;Raichoor et al. 2023;Zhou et al. 2023).The parent catalog used for DESI target selections is constructed from Data Release 9 of the DESI Legacy Imaging Surveys (Zou et al. 2017;Dey et al. 2019;Schlegel et al. 2023).The photometric data contains three optical bands grz from the DECam Legacy Survey (DECaLS, Dey et al. 2019), the Dark Energy Survey (DES, The Dark Energy Survey Collaboration 2005), the Beijing-Arizona Sky Survey (BASS, Zou et al. 2017) and the Mayall zband Legacy Survey (MzLS), and two infrared bands W 1W 2 from the Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010).Four bands grzW 1 are used to select the DESI LRG targets in 0.4 < z < 1.0 (Zhou et al. 2023) while the dashed lines denote the model predictions (not fitting).The last panel exhibits the observed number densities of LRG subsamples as well as the modeled SMF Φ(M * ) measured in simulation.The reduced χ 2 marked on the top left panel is calculated using only the statistics being fitted (see also Equation 9).To make a clear presentation, each wp has been multiplied by a factor of 3 n , where n is taken as 0, 1, 2 and 3 from the bottom one to the top one (except for the top left panel in which n changes from 0 to 2).
a grz color cut are designed to select the DESI ELG targets in 0.6 < z < 1.6 (Raichoor et al. 2023).
The One-Percent survey (also known as the 1 % survey) is the final stage of DESI Survey Validation (SV, DESI Collaboration et al. 2023a).It was operated from 5 April 2021 to 10 June 2021, covering 20 separate "rosette" areas, each of which is approximately 7 deg 2 .Thus, the total sky area of the One-Percent survey is about 1 % of the main survey.The One-Percent survey adopts the same observing mode as the main survey, but it performs spectroscopic measurements for all potential targets as much as possible by conducting many repeated visits.Because of the high fiber-assignment and spectroscopic rate, the galaxy samples in the One-Percent survey are nearly complete.The Sky coverage of the One-Percent survey is displayed in Figure 1.
We use the One-Percent survey LSS clustering catalog, which is part of the DESI Early Data Release (EDR, DESI Collaboration et al. 2023b).The LSS catalog contains all target classes with successful redshift measure-ments from the internal "Fuji" spectroscopic data releases.The completeness weight for each galaxy is estimated by performing 128 alternative Merged Target List (MTL) realizations.The assignment probability PROB of a target can be calculated by N assigned /N tot , where N tot = 129 is the total number of realizations and N assigned indicates the number of times a target is assigned in these 129 realizations.Then the completeness weight w c is defined as 129/(128 × PROB + 1).By comparing the ELG auto correlation functions with and without angular upweighting (Mohammad et al. 2020), we find that this weighting can boost the w p by about 5% at r p ∼ 0.5 Mpc h −1 and by about 10% at r p ∼ 0.1 Mpc h −1 .Given the relatively large uncertainty of the ELG auto correlations on small scales, the effect of the angular upweighting on our results is negligible.

LRG and ELG samples
Taking the photometry data in five bands grzW 1W 2, we perform spectral energy distribution (SED) fitting for the LRG and ELG samples using CIGALE (Boquien  The reduced χ 2 marked on the top left panel is calculated using only the statistics being fitted (see also Equation 10).et al. 2019).We adopt the stellar spectral library provided by Bruzual & Charlot (2003) to construct stellar population synthesis models.The initial stellar mass function (IMF) provided by Chabrier (2003) is used in the calculation.We suppose three different metallicities Z/Z = 0.4, 1.0, 2.5 in our model.A delayed star formation history (SFH) φ(t) t exp(−t/τ ) is assumed, where the timescale τ spans from 10 7 to 1.258 × 10 10 yr with an equal logarithmic space of 0.1 dex.We apply the starburst reddening law of Calzetti et al. (2000) to calculate the dust attenuation, in which the color excess E(B − V ) varies from 0 to 0.5.
We calculate the number density of galaxies as functions of the stellar mass and redshift, which are presented in Figure 2.They are the stellar mass functions (SMFs) of the observed samples.One should be aware that the observed SMFs may suffer from various target selections, and they should be distinguished from the intrinsic ones.Nevertheless, the figure tells us the basic properties of the samples.First, for the LRG sample, its SMF hardly changes between redshifts 0.4 and 1.0 at the massive end M * > 10 11.3 M .This is also consistent with previous studies (Pozzetti et al. 2007(Pozzetti et al. , 2010;;Davidzon et al. 2013;Xu et al. 2022aXu et al. , 2023)).At the massive end, Xu et al. (2023) demonstrated that the SMF has nearly no evolution since z < 0.7.Here we consider the massive LRG with M * > 10 11.3 M as a stellar mass-complete sample.At the low mass end (M * < 10 10.3 M ), the SMF of LRG shows a second peak.We find that these galaxies near this peak are dusty star-forming galaxies with large E(B − V ) (about 0.4-0.5).We have also looked at the individual 5-band magnitudes, and found that most of these galaxies are too bright in W 2 relative to the LRG SED expectation, which also prefers dusty star-forming SEDs.Moreover, this population is rather small (< 1 %) and mostly at low redshifts (0.4 < z < 0.6), therefore our results in this paper are not affected by these galaxies.
As for the ELG, the change in the SMF is more remarkable from z = 0.6 to z = 1.6.The number density of ELG reaches a maximum at z ∼ 1 and then decreases with increasing redshift.At fixed redshift, the SMF of ELG shows a peak between 10 9 M and 10 10 M .On the one hand, the ELG targets in DESI are mainly blue galaxies with intense star formation and strong emission lines.Massive galaxies are usually more prone to quenching and reddening, resulting in a decrease in the ELG number density at the massive end.On the other hand, some low-mass faint galaxies could be excluded in target selection, leading to a drop in the SMF at the low-mass end.Nevertheless, due to the low mass-to-light ratio of these blue galaxies, there is still a considerable  The reduced χ 2 marked on the top left panel is calculated using all the statistics being fitted (see also Equation 12).number of ELGs even at 10 8.5 M .The changes of the ELG SMF with redshift also reflect the complexity of its target selection.Combining the massive LRGs and the low-mass ELGs, our galaxy samples can cover a wide range of stellar mass.
In addition to stellar mass, we also present the distribution of redshift and [O II] luminosity for the ELG samples in Figure 3.The [O II] fluxes are taken from the EDAv1 catalog.We notice that some bright ELGs are missing at z < 0.8, which may be attributed to the bright cut of g > 20 mag in the ELG target selection (Raichoor et al. 2023).This cut intends to reduce the contamination of galaxies at low redshifts.Therefore, we first focus on the redshift range 0.8 < z ≤ 1.0, where the ELG has a more complete population and achieves the highest number density, and the LRG sample is also complete.
To investigate the cross correlation between LRG and ELG, we divide our LRG and ELG samples at 0.8 < z ≤ 1.0 into different subsamples (LRG0, LRG1, LRG2, LRG3, ELG0, ELG1, ELG2, ELG3) binned by stellar mass.The number of galaxies and the stellar mass range of each subsample are presented in Table 1 and 2. It should be emphasized that LRG0 may be somewhat incomplete in stellar mass, so we only use its auto and cross correlation functions instead of its number density in the subsequent fitting process.Besides, since the number of galaxies in LRG3 is limited, its clustering signal is severely affected by Poisson noise.We do not include the correlation functions of LRG3 in our fitting.

Estimation of galaxy correlation function
For subsamples x and y, their two-dimensional correlation function can be estimated via the Landy-Szalay estimator (Landy & Szalay 1993;Szapudi & Szalay 1998) where r p and r π correspond to the two components that are perpendicular and parallel to the line-of-sight, and D x D y , D x R y , D y R x and R x R y are the normalized weighted pair counts for galaxy-galaxy, galaxy-random, random-galaxy and random-random, respectively.A total of 20 equally logarithmic r p bins from 0.1 to 30 Mpc h −1 and 40 equally linear r π bins from 0 to 40 Mpc h −1 are set in the measurement.We use all the random samples provided by the One-Percent survey catalog to account for the survey geometry.We keep the RA and Dec of these random samples but assign them new redshifts by shuffling the observed redshifts of each subsample.
To obtain the real-space projected correlation function w p,xy (r p ), we integrate ξ xy (r p , r π ) along the lineof-sight direction (Davis & Peebles 1983) by where r π,max = 40 Mpc h −1 .Utilizing the jackknife technique, we can quantify the covariance matrix of the observed correlation functions.We divide the survey areas of One-Percent survey into 100 approximately equal small fields according to the distribution of random points.The covariance matrix is computed by where N jack = 100 and w k p,i(j) is measured from the k-th jackknife region, here i (j) represents the i (j)-th r p bin.
The observed w p are presented as data points with error bars in Figure 4.
Similar to the measurement of ξ xy (r p , r π ), we also calculate the correlation functions ξ xy (s, µ) in redshiftspace and express them as multipole moments (Hamilton 1992) where L l (µ) is the Legendre function, and l can be specified as 0 (monopole), 2 (quadrupole) and 4 (hexadecapole).We take 15 equally logarithmic s bins from 0.3 to 30 Mpc h −1 and 10 equally linear µ bins from 0 to 1 in the measurements.

MODELING
In this section, we propose a concise abundance matching technique to connect the observed LRG and ELG samples to dark matter halos in N-body simulation.

N-body simulation
We adopt the CosmicGrowth (Jing 2019) simulation suite in this study.The CosmicGrowth is performed by an adaptive parallel P 3 M algorithm (Jing & Suto 2002), and releases a series of N-body simulations with different cosmological parameters and different resolutions.We choose one of the ΛCDM simulations of CosmicGrowth to model the galaxy-halo connection.This simulation adopts the standard cosmology: Ω m = 0.268, Ω Λ = 0.732, h = 0.71, n s = 0.968 and σ 8 = 0.83, and has a total of 3072 3 dark matter particles in a 600 Mpc h −1 box, which yields a high mass resolution of m p = 5.54 × 10 8 M h −1 .The halo groups are identified by the friends-of-friends algorithm (FOF) (Davis et al. 1985) with a linking length of b = 0.2, while the subhalo merger trees are built using the Hierarchical-Bound-Tracing algorithm (HBT+) (Han et al. 2012(Han et al. , 2018)).
For the subhalos with less than 20 particles, we calculate their merger time scales using the fitting formula provided by Jiang et al. (2008) and exclude those subhalos that have fully merged with their central subhalos.The halo mass function from Jing (2019) and the subhalo mass function from Xu et al. (2022b) have verified that the halos in this simulation can be well resolved to 10 10 M h −1 (about 20 particles), which is sufficient for modeling ELGs that are more likely to live in lowmass halos (e.g., Favole et al. 2016;Guo et al. 2019;Hadzhiyska et al. 2021;Okumura et al. 2021).The mass of a host halo M h is defined as its current M vir that is the mass enclosed by a virialized spherical structure with an over-density ∆ vir (z) (Gunn & Gott 1972;Bryan & Norman 1998).The accretion mass of a subhalo M s is defined as the M vir at the snapshot before it was accreted by the current host halo.Moreover, for the calculation of clustering, we define the z-axis as the direction of the line-of-sight and add the RSD effects to the coordinate components along this direction for all the halos and subhalos in simulation.To cover the whole redshift range of ELGs in the observation (0.6 < z ≤ 1.6), we take a total of five halo catalogs from snapshots at z = 0.71, 0.92, 1.09, 1.27 and 1.47.

Stellar-halo mass relation
To link the dark matter halos in the simulation to the observed galaxies, we use stellar mass as a bridge.For a given halo (subhalo) with mass M h , we assume that the stellar mass of the hosted galaxy follows a Gaussian conditional probability distribution function (PDF) (5) where M * |M h denotes the mean SHMR and σ is the scatter of this relation.We adopt a double power-law function (Wang et al. 2006;Wang & Jing 2010;Yang et al. 2012;Moster et al. 2013) to parameterize the where M 0 divides the SHMR into two parts with different slopes α and β, and k is a normalization constant.
Similarly, the relation between stellar mass and accretion mass p(M * |M s ) for a subhalo can also be established using the above model.Since the stellar mass of a galaxy can still be influenced by the subsequent evolution process after infalling (Yang et al. 2012), the SHMR of a subhalo should be expected to be different from that of a halo.However, this difference is small in the modeling of galaxy clustering (Wang & Jing 2010), especially when the observed sample size is limited.Therefore, similar to other works (e.g., Wang & Jing 2010;Behroozi et al. 2019;Xu et al. 2022bXu et al. , 2023)), we adopt a unified model with five parameters: {α, β, M 0 , k, σ} to describe the p(M * |M h ) for halos and p(M * |M s ) for subhalos.After the release of DESI Y1 data, we will attempt to construct the SHMR for halos and subhalos separately and test this assumption.

Modeling LRGs and ELGs in simulation
Once the SHMR is established, we can assign stellar mass to each halo and subhalo.In this way, we can obtain a population of galaxies with complete stellar masses in the simulation.We refer to this kind of galaxy as a normal galaxy.
Firstly, from the normal galaxies, we can directly select the LRG subsamples (LRG0, LRG1, LRG2, LGR3) according to the ranges of their stellar mass bins.As mentioned in Section 2.2, compared to LRG1, LRG2 and LRG3, the LRG0 in the observation may be somewhat incomplete.Although we still choose all normal galaxies with M * between 10 11.1 and 10 11.3 M as LRG0, we only consider their clustering rather than their number density.
Next, we improve the approach provided by Paper I to select ELGs.Paper I introduced a constant parameter f sat to reduce the fraction of satellite galaxies in the simulation, and then randomly sampled ELGs from the simulated normal galaxies based on the ELG number density in observation.Here, we further divide this process into two steps.
The first step is to select ELG candidates from the normal galaxies.We keep all central galaxies in the normal population as ELG candidates, that is, the probability of the central galaxies to be selected is P cen = 1.Since satellite galaxies are more likely to be quenched by the environment, the probability that a satellite galaxy becomes an ELG candidate, denoted by P sat , is usually less than 1.P sat is determined by best-fitting the auto and cross correlations of ELGs and LRGs, as studied in the next section.Here in the selection of ELG candidates, we keep all central galaxies and randomly remove some satellite galaxies based on their P sat .
The second step is to select true ELGs from these ELG candidates.For a given stellar mass bin, we can measure the ELG number density nobs (M * ) in observation and the candidate number density ncan (M * ) in simulation.Then, the probability that a candidate is selected as an ELG is written as We calculate F ELG (M * ) from 10 7.5 to 10 12 M with a width ∆ log M * = 0.1, and linearly interpolate the F ELG − log M * relation to make it continuous.In this way, we can assign a probability F ELG to each candidate and randomly select ELGs based on their F ELG .Finally, the selected ELGs in simulation can be further categorized into four subsamples: ELG0, ELG1, ELG2 and ELG3.We summarize the modeling process for LRGs and ELGs in simulation as follows: (i) Populate halos and subhalos with normal galaxies according to the SHMR model p(M * |M h ) and p(M * |M s ).
(ii) Select the four LRG subsamples LRG0, LRG1, LRG2 and LRG3 from the normal galaxies population, and compute their number density n mod .
(iii) Given the probability P sat for satellite galaxies, keep all central galaxies and randomly select some satellite galaxies as ELG candidates based on their P sat .
(iv) Calculate the probability F ELG for each ELG candidate and randomly select some candidates as ELGs based on their F ELG .
(vi) Calculate the auto and cross correlation functions for all the LRG and ELG subsamples in simulation.
For each set of parameters of the SHMR and P sat model, we follow the above steps to calculate the number densities, and auto and cross correlations of LRGs and ELGs.In order to make the model predictions stable, we implement the above steps 10 times and average the results from the 10 realizations as the final w mod p and n mod .The model parameters are determined by best-fitting the model predictions to the observations, as described in the next section.3) for the constant Psat model using either ELG auto or LRGxELG cross correlation are displayed as orange and brown curves, respectively.The blue solid lines represent the best-fit result (the last column of Table 3) for the halo mass-dependent Psat model using all the correlation functions.The shallow regions denote the 1σ scatter.The horizontal gray line indicates the lowest stellar mass limit that can be probed by the ELG subsample.

FITTING PROCESS AND RESULTS
In this Section, we fit our model to the projected correlation functions and galaxy number densities.We comprehensively compare the performance of different P sat models.Using the best-fit parameters, we further predict the correlation functions in redshift-space.We also check if our model is extendable to higher redshifts.

A constant P sat model
Given the current measurements, we need to check whether a constant P sat model is sufficient to fit the observations that involve a total of 3 LRG auto correlation functions w obs p,LRG , 4 ELG auto correlation functions w obs p,ELG , 16 LRGxELG cross correlation functions w obs p,LRGxELG and 3 number densities n obs LRG of the LRG subsamples.As we will show shortly, we find that the constant P sat model is difficult to fit all the correlation functions simultaneously.Good fits to the ELG auto and LRGxELG cross correlations require very different values of P sat .Therefore, we will discuss the two cases separately.

Fitting without cross correlations
To constrain our model parameters, we first use the LRG and ELG auto correlations w obs p,LRG and w obs p,ELG , and the LRG number densities n obs LRG .For subsample x and y, we can write their χ The dashed lines are the direct model predictions using the best-fit parameters.To make a clear presentation, each sξ0 has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom to the top one (except for the top left panel in which n changes from 0 to 2).The deviation ∆ξ0 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.can be computed as where x ∈ [LRG0, LRG1, LRG2], y ∈ [ELG0, ELG1, ELG2, ELG3] and t ∈ [LRG1, LRG2, LRG3].The n obs and n mod are the observed and modeled number densities for the LRG subsamples respectively, and the observational uncertainty σ n is estimated from the fieldto-field variations of 100 jackknife fields.We explore the posterior probability distribution of the model parame-ters based on Bayesian theory, in which the logarithmic likelihood function is proportional to −0.5χ 2 .The priors for the six parameters are set as 10 < log M 0 < 13, 0.1 < α < 0.5, 1 < β < 5, 9 < log k < 12, 0 < σ < 1 and 0 < P sat < 1.Using the code emcee (Foreman-Mackey et al. 2013) for Markov Chain Monte Carlo (MCMC) analysis, we run 72 chains each with 1000 steps to sample the entire parameter space.The first 10% steps of each chain are discarded as burn-in.The best-fit models for w mod p,LRG and w mod p,ELG are shown in Figure 4 as solid lines with distinct colors.The modeled SMF is displayed in the last panel of Figure 4. We   8, but we show the observed quadrupole ξ2(s) in redshift-space and model predictions.To make a clear presentation, each s 2 ξ2(s) has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom one to the top one (except for the top left panel in which each s 2 ξ2(s) has been added with n × 100 where n changes from 0 to 1).Here we omit the ξ2(s) of the LRG2 auto correlations, because its measurement is too noisy.The deviation ∆ξ2 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.3. We can notice that the constraints on the slope β is poor.This implies that it is hard to break the parameter degeneracy of SHMR and P sat using only the w obs p,ELG .Moreover, due to the significantly high amplitude of w obs p,ELG on small scale, the best-fit P sat parameter tends to be 1.As for the LRGx-ELG cross correlations, large P sat makes the model sys-tematically overestimate the observed w obs p,LRGxELG on small scales.

Fitting without ELG auto correlations
Instead of using w obs p,ELG , we take the cross correlations w obs p,LRGxELG to fit the model.Similarly, we can write the total χ 2 as  8, but we show the observed hexadecapole ξ4(s) in redshift-space and the model predictions.To make a clear presentation, each s 2 ξ4(s) has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom one to the top one (except for the top left panel in which each s 2 ξ4(s) has been added with n × 100 where n changes from 0 to 1).Here we omit the ξ4(s) of the LRG2 auto correlations, because its measurement is too noisy.The deviation ∆ξ4 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.
fit results and the parameter constraints are presented in Figure 5 (see also Figure 15 of Appendix A) and the second column of Table 3.The model provides a suitable fit for the w obs p,LRG and the w obs p,LRGxELG and results in better constraints on the parameters of the SHMR model.However, the value of P sat is only 0.15 in this case.The predicted w mod p,ELG are obviously underestimated at r p < 1 Mpc h −1 .

A halo mass-dependent P sat model
As argued in Section 4.1, the best fitting with cross or auto correlations of the LRGs and ELGs can yield very different values of P sat .The lower amplitude of the one-halo term of w obs p,LRGxELG implies that there are few ELG satellite galaxies around the massive LRGs.This is consistent with many studies suggesting that the quenched fraction of satellites around massive central galaxies is high (Donnari et al. 2021;Zheng et al. 2023).However, the situation is reversed for the w obs p,ELG where the significantly higher clustering signal indicates that there should be more galaxy pairs at r p < 1 Mpc h −1 .Actually, this feature is also reflected in Paper I, although the measurements there had larger uncertainties.In Figure 7 of Paper I, a constant satellite fraction is sufficient to reproduce the auto correlations of the first two ELG  p,ELG for ELG subsamples.Except for the solid lines in the first panel, which are the best-fit results at 0.8 < z ≤ 1.0, all dashed lines in the other panels are the predictions by the model.To make a clear presentation, each wp,ELG has been multiplied by a factor of 3 n , where n is taken as 0, 1, 2 and 3 from the bottom to the top one.
subsamples with moderate [O II] luminosity.But for the two brightest ELG subsamples whose properties are closer to the DESI samples, the model underestimates the auto correlation on small scales.Since the sample size of VIPERS is small, Paper I was unable to perform more careful analyses.It is worth mentioning that the angular correlations of [O II] emitters at z > 1 measured by Okumura et al. (2021) also show a sudden increase at small scales.
In order to maintain a sufficient number of satellite ELGs in low-mass halos while reasonably considering the quenching effect of satellite galaxies in massive halos, we propose a halo mass-dependent P sat model as where erf is the error function and M h is the host halo mass of satellite galaxies.In this model, P sat tends to be a constant a at log M h < b, while decreases as the halo mass increases at log M h > b and finally reaches a constant c.As shown in Section 4.1.1(see also Figure 14), due to the high amplitude of w obs p,ELG at small scales, the best-fit value of P sat converges to 1 for the ELG satellite galaxies in low-mass halos.We therefore fix the parameter a = 1 in Equation 11.Combining all the measurements, the total χ 2 can be computed as ].The fitting results for this model are displayed in Figure 6, Figure 16 of Appendix A and the third column of Table 3.We present all the best-fit SHMR and P sat models in Figure 7.
In general, the reduced χ 2 /dof = 0.89 indicates that the overall fit is reasonable.The model can reproduce both the w obs p,LRGxELG at r p > 0.1 Mpc h −1 and the w obs p,ELG at r p > 0.3 Mpc h −1 , and thus overcomes the shortcomings of the constant P sat model.The five parameters of the unified SHMR model can be well constrained.This illustrates that the combination of massive LRG samples and low-mass ELG samples can effec-

Gao et al.
tively help us to determine the SHMR in a wide range of stellar mass.
In the halo mass-dependent P sat model, the parameter a, which represents ELG satellite probability in small halos, is set to 1.However, such a P sat value is still difficult to reproduce the one-halo term of w obs p,ELG at r p < 0.3 Mpc h −1 , which is still higher.One possible explanation is 1-halo galaxy conformity (e.g., Weinmann et al. 2006;Calderon et al. 2018;Zu et al. 2022;Hadzhiyska et al. 2022a), which describes a phenomenon that the physical properties of central and satellite galaxies in the same halo are correlated.There is still a debate about whether galaxy conformity exists.By comparing the SDSS observations to mock catalogs, Calderon et al. (2018) argued that the 1-halo conformity is not real and could be caused by group-finding systematics.Using the abundance and weak lensing measurements of SDSS clusters, Zu et al. (2022) detected a halo massdependent galaxy conformity between the stellar mass of bright central galaxies (BCGs) and the cluster satellite richness.Hadzhiyska et al. (2022a) have shown that if a halo already contains an ELG satellite, its central galaxy is twice as likely to be an ELG.They further introduce a free parameter in the HOD to interpret this conformity effect.Recently, Rocher et al. (2023) also introduce a parameterized conformity bias in the HOD model to reproduce the auto correlation of ELGs in the One-Percent survey.Nonetheless, the conformity of ELGs may have more complicated dependencies on different physical properties such as stellar mass, emission line strength and environment.In our future work, we will develop a comprehensive model to quantify the conformity of ELGs.

Checking the correlation functions in redshift-space
Using the best-fit SHMR and P sat models, we make a prediction for the multiple moments ξ 0 (s), ξ 2 (s) and ξ 4 (s) in redshift-space.In order to make a fair comparison with the observations, we have incorporated redshift measurement errors and galaxy velocity bias in our model.With multiple independent spectroscopic observations for the same object, Lan et al. (2023) have presented the uncertainty distributions of the redshifts for BGSs, LRGs and ELGs in DESI.Accordingly, we assume that the redshift uncertainty follows a Gaussian distribution with a scatter of σ z , where σ z is fixed to 40 km s −1 and 10 km s −1 for the modeled LRGs and ELGs respectively.Considering that a central galaxy is typically not at rest with respect to its host halo (Yoshikawa et al. 2003), we also randomly assign a Gaussian distribution with σ c = α c × σ v for each halo to account for the 1-D velocity bias.Here α c is set as 0.22 that is determined by Guo et al. (2015) using BOSS CMASS galaxies, while σ v = (GM vir /(2R vir )) 0.5 is the 1-D velocity dispersion of the halo.
The observed multiple moments and our model predictions are displayed in Figure 8, 9 and 10.Here we omit the ξ 2 (s) and ξ 4 (s) of the auto correlations of LRG2 due to their too noisy measurements.We show that although we use only w obs p in fitting, our model can well reproduce the auto correlations of LRGs and the cross correlations of LRGsxELGs in redshift-space.Similar to the case in real-space, for the auto correlations of ELGs, the predicted ξ 0 (s), ξ 2 (s) and ξ 4 (s) are underestimated on small scales.Nevertheless, at s > 1 Mpc h −1 where the Kaiser effect plays a dominant role, our model is sufficient to reproduce the observed multiple moments.

Extending the ELG-halo connection to higher redshifts
Since the overlap between LRGs and ELGs is only in the redshift range of 0.8 < z < 1.0, beyond which we cannot constrain the ELG-halo connection by their cross correlations.We need to check whether our model can be safely extended to higher redshifts.
We divide all the ELG samples within 0.8 < z ≤ 1.6 into four redshift bins and measure their auto correlations.However, for redshift 0.6 < z ≤ 0.8, due to the limited number of ELG samples and the incompleteness at the bright end (see also Figure 3), we do not consider this redshift range here.Using four individual snapshots at redshifts 0.92, 1.09, 1.27 and 1.47, we calculate the modeled w mod p for each redshift bin using the best-fit parameters presented in Section 4.2.We exhibit the evolution of w obs p,ELG as well as the model predictions in Figure 11.At redshift 1.0 < z ≤ 1.6, the model predictions can match well the measurements at r p > 0.5 Mpc h −1 .This implies that the SHMR and the ELGhalo connection evolve weakly from redshift z = 0.8 to 1.6.Although our model is obtained at z ∼ 0.9 and has only seven free parameters, it can still reproduce the observations over the entire range of 0.8 < z < 1.6 given the current measurement errors.This will allow us to build an ELG mock catalog for the DESI One-Percent and Y1 surveys.

Theoretical derivation
One of the advantages of our abundance matching approach is that we can theoretically derive the HOD of the ELGs without assuming a complicated parameterized HOD model.Given a halo with mass M h , the probability of its central galaxy becoming an ELG can be calculated as (13) where p(M * |M h ) is the SHMR in Equation 5, and F ELG (M * ) = nobs (M * ) /n can (M * ) is the observed ELG fraction in the modeled ELG candidates as defined in Equation 7. Similarly, given a subhalo with mass M s , the probability of the satellite galaxy in it becoming an ELG can be written as where we select the halo mass-dependent P sat (M h ) model shown in Equation 11 as the P sat (M h ).
The key step is to compute the number density of ELG candidates in the model and derive F ELG (M * ).Firstly, the total number density ncan (M * ) of the ELG candidates can be written as where ncan cen (M * ) and ncan sat (M * ) are the number densities of the central and satellite ELG candidates, respectively.Then, ncan cen (M * ) can be calculated via where P (M * |M h ) is the probability that a halo hosts a central galaxy with stellar masses between log M * − ∆ log M * /2 and log M * + ∆ log M * /2, and n(M h ) is the halo mass function.To numerically evaluate this integration, from 10 10 to 10 15 M h −1 , we divide the halo samples into 50 tiny bins in logarithmic space with a width of ∆ log M h = 0.1.For the i-th bin, we can calculate the probability P (M * |M h,i ) by and measure n(M h,i ) directly from simulation.Analogously, the number density of satellite candidates ncan sat (M * ) can also be derived in this way where subhalo mass function ns (M s,j |M h,i ) is measured in 50 × 50 grids, and the probability P (M * |M s,j ) for the j-th subhalo mass bin can be computed via Finally, the HOD forms of ELGs can be expressed as follows (20) Using the above formula, we can obtain the central and satellite occupation numbers of ELGs for a given halo with mass M h .
In Figure 12, we present the theoretical HODs calculated by Equation 20 as solid lines.To further make a self-consistent test, we perform 10 random realizations to populate ELGs in simulations and directly measure their HODs.The average results from the 10 realizations are shown as dots in Figure 12.
The central occupations of our model can clearly decompose ELGs into two different populations.On the one hand, at the low-mass end, N cen (M h ) shows a similar peak at 10 11.5 to 10 12 M h −1 , which is consistent with other studies (e.g., Favole et al. 2016;Guo et al. 2019;Hadzhiyska et al. 2021;Okumura et al. 2021).As the redshift increases, the location of the peak gradually shifts towards the high-mass end.It may be due to the fact that the selected ELG samples in DESI have relatively higher stellar mass at higher redshift (see also the right panel of Figure 2) and hence tend to reside in larger halos.Beyond the peaks, N cen (M h ) begins to decay from 10 12 to 10 13 M h −1 , which reflects the fact that star formation in massive galaxies has almost been stopped (Xu et al. 2020).However, on the other hand, there is an upturn in N cen (M h ) at about M h > 10 13.5 M h −1 .This may actually correspond to another population of ELGs.Central galaxies in massive halos are more likely to host AGN, and the high-energy radiation from their central engine can ionize the surrounding gas and produce strong emission lines (Comparat et al. 2013).Therefore, although lowmass star-forming ELGs dominate our current sample, we also need to pay more attention to those massive ELGs that may be related to AGN activities.
For the satellite occupation N sat (M h ), its shape is mainly determined by the P sat (M h ) model.At M h < 10 12 M h −1 where P sat is close to 1, N sat increases rapidly with the host halo mass in a power-law form.Then, as P sat decreases with M h at 10 12 < M h < 10 14 M h −1 , N sat keeps almost a constant because the increase in the number of subhalos with M h roughly cancels out the decrease of P sat .Above M h > 10 14 M h −1 where P sat stays at a constant about 0.04, N sat is nearly proportional to the host halo mass M h , as the number of subhalos within a host halo is approximately proportional to M h (e.g., Gao et al. 2004Gao et al. , 2012;;Giocoli et al. 2008;Han et al. 2016).
From the derived HOD, we find that the HOD of ELGs depends on the host halo mass and the observed redshift in a complicated way.It would be difficult to find an analytical form to represent the HOD of ELGs at different redshifts.And the analytical models at different redshifts may have different sets of parameters.It is worth mentioning that although our model is obtained from the SHAM method, our derived HOD forms can still be applied to populating ELGs with halos in low-resolution simulations where subhalos are not resolved.Therefore, we believe that the SHAM approach presented here has advantages over the conventional HOD method, and can be easily applied to N-body simulations to produce mock catalogs for ELGs.

Mean host halo mass and satellite fraction of
ELGs in the model For comparison with other HOD studies, we calculate the mean host halo mass M h of ELGs and the ELG satellite fraction f sat in the model.Based on the derived HODs at different redshifts, we can calculate the average mass M h of the host halos for ELGs: where ) is the total occupation number of ELGs.We present the mean halo mass M h as function of redshifts in the left panel of Figure 13.Similarly, we also calculate the mean halo mass for central ELGs M h cen and satellite ELGs M h sat , and show them in Figure 13.We can see that the central ELGs tend to occupy larger halos at higher redshift, while the host halo mass of the satellite ELGs decrease with redshift.But the overall M h is almost a constant across the probed redshift range.Rocher et al. (2023) also show that the evolution of the host halo mass of ELG with redshift is weak ( log M h ∼ 11.8 at both 0.8 < z < 1.1 and 1.1 < z < 1.6), although their values are slightly lower.We can further compute the mean ELG satellite fraction f sat in the model: The f sat at each redshift is shown in the middle panel of Figure 13.It shows a monotonically decreasing evolution with increasing redshift.This trend is similar to the study of Guo et al. (2019) in which the f sat of the eBOSS ELG samples also decreases with redshift.In addition, since the P sat (M h ) depends on the halo mass, the ELG satellite fraction in our model is not a constant.Here we also represent f sat (M * ) as a function of stellar mass at each redshift, and show them in the right panel of Figure 13.All the values of M h and f sat are listed in Table 4.

DISCUSSION
There is a series of works in parallel that use different methods for constructing the galaxy-halo connection for LRGs and ELGs in the One-percent survey (e.g., Yuan et al. 2023;Ereza et al. 2023;Rocher et al. 2023;Lasker et al. 2023;Yuan et al. 2023;Yu et al. 2023;Prada et al. 2023).
For example, Yuan et al. ( 2023) perform a comprehensive HOD analysis for LRGs and QSOs using AbacusSummit simulation (Maksimova et al. 2021).They combine the standard HOD model with incompleteness parameter, galaxy velocity bias, and galaxy assembly bias.They test these possible HOD extensions by fitting LRG correlation functions at different redshifts.The best-fit values of the velocity bias are mostly consistent with the previous BOSS results.They demonstrate that the galaxy assembly bias has almost no effect on the fitting given the current measurement precision.
For the HOD modeling of ELGs, Rocher et al. ( 2023) explore the performance of four different HOD forms in the fitting.In particular, to recover the strong clustering of ELGs on small scales, they introduce a strict central-satellite conformity bias that only allows satellite pairs to exist in the halos whose central galaxy is an ELG.They also argue that the velocity dispersion of ELG satellites should be larger than that of the dark matter particles.Yu et al. (2023) develop a generalized SHAM approach to model the LRGs, ELGs and QSOs in UNIT simulation (Chuang et al. 2019).Besides the intrinsic dispersion of the galaxy-halo relation, they further model the uncertainty of the redshift measurement and add an incompleteness parameter to account for the absence of galaxies in the massive halos.They modeled the clustering only at large scale s > 5 Mpc h −1 .In particular for ELGs, they show that the ELG satellite fraction f sat in the model should be suppressed to ∼ 4% to recover the observed redshift clustering.
Using Uchuu simulations (Ishiyama et al. 2021), Prada et al. (2023) generate lightcone mocks for four DESI tracers through the SHAM method.For the LRGs, they first adopt a complete SMF from the PRIMUS survey to implement the SHAM process, and then down-sample the galaxies in simulation based on the observed LRG SMF.They find a systematically low clustering of the mocks below ∼ 5 Mpc h −1 , probably due to the different properties of the PRIMUS and DESI samples.For the ELGs, they assume that the maximum circular velocity of the host halos (subhalos) is a Gaussian distribution with its amplitude normalized to the observed number density.In their model, the ELG satellite fraction f sat is considered as a free parameter.The observed ELG clustering can be reproduced by their model at s > 4 Mpc h −1 .
Each of the above studies has its pros and cons.In comparison, the main advantages of our method can be summarized as follows.(1) We have used the auto and cross correlations of ELGs and LRGs jointly to simultaneously constrain the SHMR (SMF) and the ELGhalo relation.The former can be used to generate LRG mocks.
(2) Because we model ELGs and LRGs jointly, we demonstrate that a mass-dependent P sat model is essential to describe satellite ELGs in both small and large halos.(3) With our SHAM model, we can derive the conventional HOD which can also be applied to coarse simulations.(4) Given the current measurement uncertainty, our model for ELGs can be extended to the entire redshift range of 0.8 < z < 1.6, without introducing a large number of parameters at each redshift.Our model is accurate in describing the clustering of LRGs and ELGs down to sub Mpc h −1 scales.

SUMMARY
In this work, we extend our novel method (Gao et al. 2022) to accurately construct galaxy-halo connections for LRGs and ELGs using the DESI One-Percent survey.Our method can simultaneously constrain the SHMR for normal galaxies and in particular the ELG-halo connection in the One-Percent survey.We summarize our main results as follows.
1. Using the galaxy catalog from DESI One-Percent survey, we perform a SED fitting and measure the apparent SMFs for the LRG and ELG samples.At redshift 0.8 < z ≤ 1.0, we divide the LRG and ELG samples into eight subsamples based on their stellar masses.We estimate the LRG auto correlations, ELG auto correlations and LRGxELG cross correlations for all galaxy subsamples.
2. Combining the abundance matching technique and a high-resolution N-body simulation from CosmicGrowth (Jing 2019), we simultaneously model the galaxy clustering for LRGs and ELGs.
We adopt the SHMR model proposed by Wang & Jing (2010) to establish the normal galaxy-halo re-lation.Given the SHMR, normal galaxies can be populated to halos and subhalos in the simulation.We select stellar mass-complete LRG samples from the massive normal galaxies.We select ELGs in two steps.We first consider all central galaxies as ELG candidates, while reducing the probability that satellite galaxies become ELG candidates with the adjusting parameter P sat .Then, we calculate the ELG fraction F ELG (M * ), which is the ratio of the number density of ELGs in the observation to that of the ELG candidates in the model.By assigning a probability F ELG (M * ) to each candidate, the ELG samples can be randomly selected from these candidates.
3. We utilize MCMC analysis to explore the parameter space of our model.With the LRG samples, the massive end of SHMR can be well determined, while the ELG samples provide much information for the SHMR down to 10 8.5 M .We also do a test for the different P sat models.We find that the ELG auto correlations and the LRGx-ELG cross correlations can lead to very different P sat values.Thus, we propose a host halo massdependent P sat model.This model can reasonably reduce the P sat for the massive halos while retaining a sufficient number of satellite ELGs in small halos.Our model can well reproduce the SMF of LRG at the massive end, the LRG auto correlations, the LRGxELG cross correlations and the ELG auto correlations at r p > 0.3 Mpc h −1 .Using this model, we further predict the multiple moments in redshift-space.Although our model has only seven parameters that are fully fixed by the projected correlations at z ∼ 0.9, our model predictions are consistent with the redshift-space correlatins at s > 1 Mpc h −1 .We also check if our model is valid for other redshifts covered by DESI, and we find that our model can match well the ELG auto correlations at r p > 0.5 Mpc h −1 for the entire redshift range 0.8 < z ≤ 1.6.Thus our model can be used for generating ELG mock catalogs.
4. Based on our model, we theoretically derive the HOD forms for ELGs in the One-Percent survey at different redshifts.The shape of the central occupations indicates that ELGs can be plainly divided into two populations.Star-forming galaxies dominate the low-mass end of ELGs, which tend to reside in halos with mass 10 11.5 to 10 13 M h −1 .We have seen the upturn of the HOD at halo mass > 10 13 M h −1 , which might indicate that AGN galaxies may contribute to the massive end of the ELG SMF.Overall, the HODs of both central and satellite galaxies show a complicated dependence on host halo mass and redshift, which may indicate that it is challenging to find simple analytical forms to represent the HOD of ELGs in the DESI survey and/or future ELG surveys.Our model can describe the evolving HODs of ELGs with just seven parameters, without introducing different sets of parameters at different redshifts.
In our subsequent work, we will investigate the origin of the strong clustering of ELGs on very small scales (less than a few hundred kpc h −1 ) and further refine our model.Meanwhile, we will also generate realistic ELG lightcone mock catalogs for the DESI One-Percent and Y1 surveys based on our model, and study its impact on future DESI cosmological probes such as BAOs, redshift distortion, and weak lensing.

Figure 1 .
Figure 1.Sky coverage of the DESI One-Percent survey.

Figure 2 .Figure 3 .
Figure 2. Evolution of the SMFs of the LRG and ELG samples in the One-Percent survey.The left and right panels correspond to LRG and ELG respectively.The data points with Poisson errors denote the observed SMFs in different redshift bins.In the measurements, each galaxy has been multiplied by the completeness weight as mentioned in Section 2.1.

Figure 5 .
Figure 5. Similar to Figure 4, but fitting results for the constant Psat model with w obs p,LRG , w obs p,LRGxELG and n obs LRG .The reduced χ 2 marked on the top left panel is calculated using only the statistics being fitted (see also Equation10).

Figure 6 .
Figure 6.Similar to Figure 4 and Figure 5, but fitting results for the halo mass-dependent Psat model with all the cross (auto) correlations w obs p,LRG , w obs p,ELG and w obs p,LRGxELG , and the LRG number densities n obs LRG .The reduced χ 2 marked on the top left panel is calculated using all the statistics being fitted (see also Equation12).

Figure 7 .
Figure 7. Constraints of SHMR (left panel) and Psat (right panel) models.The best-fit results (the first two columns of Table3) for the constant Psat model using either ELG auto or LRGxELG cross correlation are displayed as orange and brown curves, respectively.The blue solid lines represent the best-fit result (the last column of Table3) for the halo mass-dependent Psat model using all the correlation functions.The shallow regions denote the 1σ scatter.The horizontal gray line indicates the lowest stellar mass limit that can be probed by the ELG subsample.

Figure 8 .
Figure8.Observed monopole ξ0(s) in redshift-space and model predictions.The data points with error bars represent the measurements.The different auto and cross correlations of LRGs and ELGs are shown in different panels.The dashed lines are the direct model predictions using the best-fit parameters.To make a clear presentation, each sξ0 has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom to the top one (except for the top left panel in which n changes from 0 to 2).The deviation ∆ξ0 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.

Figure 9 .
Figure 9. Similar to Figure8, but we show the observed quadrupole ξ2(s) in redshift-space and model predictions.To make a clear presentation, each s 2 ξ2(s) has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom one to the top one (except for the top left panel in which each s 2 ξ2(s) has been added with n × 100 where n changes from 0 to 1).Here we omit the ξ2(s) of the LRG2 auto correlations, because its measurement is too noisy.The deviation ∆ξ2 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.
also present the model predictions for w mod p,LRGxELG as dashed lines.Both w obs p,LRG and w obs p,ELG at r p > 0.1 Mpc h −1 are well fitted by this model.The posterior distributions of the model parameters are presented in Figure 14 of Appendix A. The best-fit parameters are shown in the first column of Table

Figure 10 .
Figure10.Similar to Figure8, but we show the observed hexadecapole ξ4(s) in redshift-space and the model predictions.To make a clear presentation, each s 2 ξ4(s) has been added with a constant n × 30, where n is taken as 0, 1, 2 and 3 from the bottom one to the top one (except for the top left panel in which each s 2 ξ4(s) has been added with n × 100 where n changes from 0 to 1).Here we omit the ξ4(s) of the LRG2 auto correlations, because its measurement is too noisy.The deviation ∆ξ4 of the model from the data divided by the measurement error σ is also shown at the bottom of each panel.

Figure 11 .
Figure 11.Evolution of projected auto correlation functions for the One-Percent survey ELGs.Different panels represent different redshift intervals.The data points with error bars show the measurements of w obsp,ELG for ELG subsamples.Except for the solid lines in the first panel, which are the best-fit results at 0.8 < z ≤ 1.0, all dashed lines in the other panels are the predictions by the model.To make a clear presentation, each wp,ELG has been multiplied by a factor of 3 n , where n is taken as 0, 1, 2 and 3 from the bottom to the top one.

Figure 12 .Figure 13 .
Figure 12.HODs of the ELGs in the One-Percent survey.The central occupation Ncen(M h ) and the satellite occupation Nsat(M h ) are presented in the left and right panels respectively.The solid lines denote the results of the theoretical calculations using Equation 20.The colored dots show the averaged HODs measured directly from 10 random realizations.The different colors correspond to the HODs at different redshifts.

Figure 14 .
Figure14.The posterior distributions of the parameters in the SHMR and the constant Psat fitted with w obs p,LRG , w obs p,ELG and n obs LRG .The contours show the joint distribution of each pair of parameters, and the three levels represent 68.3%, 95.4% and 99.7% confidence intervals.The median and 1σ uncertainty derived from the marginalized distributions of each parameter are presented on the top panel of each column.

Figure 15 .
Figure 15.Similar to Figure 14, but for the posterior distributions of the parameters in the SHMR and the constant Psat fitted with w obs p,LRG , w obs p,LRGxELG and n obs LRG .

Table 1 .
Details of four LRG subsamples.

Table 2 .
Details of four ELG subsamples.

Table 3 .
Best-fit parameters of the SHMRs for different Psat models.constant Psat constant Psat Psat(M h )

Table 4 .
The values of mean halo mass and satellite fraction of ELGs in our model.Note-The unit of M h , M h cen and M h sat is M h −1 .