The DESI One-Percent Survey: Exploring A Generalized SHAM for Multiple Tracers with the UNIT Simulation

We perform SubHalo Abundance Matching (SHAM) studies on UNIT simulations with \{$\sigma, V_{\rm ceil}, v_{\rm smear}$\}-SHAM and \{$\sigma, V_{\rm ceil},f_{\rm sat}$\}-SHAM. They are designed to reproduce the clustering on 5--30$\,\hmpc$ of Luminous Red Galaxies (LRGs), Emission Line Galaxies (ELGs) and Quasi-Stellar Objects (QSOs) at $0.4<z<3.5$ from DESI One Percent Survey. $V_{\rm ceil}$ is the incompleteness of the massive host (sub)haloes and is the key to the generalized SHAM. $v_{\rm smear}$ models the clustering effect of redshift uncertainties, providing measurments consistent with those from repeat observations. A free satellite fraction $f_{\rm sat}$ is necessary to reproduce the clustering of ELGs. We find ELGs present a more complex galaxy--halo mass relation than LRGs reflected in their weak constraints on $\sigma$. LRGs, QSOs and ELGs show increasing $V_{\rm ceil}$ values, corresponding to the massive galaxy incompleteness of LRGs, the quenched star formation of ELGs and the quenched black hole accretion of QSOs. For LRGs, a Gaussian $v_{\rm smear}$ presents a better profile for sub-samples at redshift bins than a Lorentzian profile used for other tracers. The impact of the statistical redshift uncertainty on ELG clustering is negligible. The best-fitting satellite fraction for DESI ELGs is around 4 per cent, lower than previous estimations for ELGs. The mean halo mass log$_{10}(\langle M_{\rm vir}\rangle)$ in $\Msun{}$ for LRGs, ELGs and QSOs are ${13.16\pm0.01}$, ${11.90\pm0.06}$ and ${12.66\pm0.45}$ respectively. Our generalized SHAM algorithms facilitate the production of mult-tracer galaxy mocks for cosmological tests.


INTRODUCTION
ΛCDM is the standard model of modern cosmology that describes the evolution of the Universe.In this framework, two dark components, dark matter and dark energy, comprise 95 per cent of the total energy density.The nature of dark energy can be explored using baryonic acoustic oscillation (BAO; Eisenstein & Hu 1998), a standard ruler of the universe.Meanwhile, redshift-space distortion (RSD, Kaiser 1987) embodies the growth rate of the large-scale structure (LSS) which is dominated by the evolution of dark matter.
The largest ongoing spectroscopic survey, the Dark Energy Spectroscopic Instrument (DESI, 2021(DESI, -2026;;DESI Collaboration et al. 2016a,b) is a robotic, fiber-fed, highly multiplexed survey that operates on the Mayall 4-meter telescope at Kitt Peak National Observatory.It aims to explore the nature of dark energy via the most precise measurement of the 3D Universe in 14, 000 deg 2 of the sky after 5 years of observations (Levi et al. 2013).The list of targets to be observed by DESI (Myers et al. 2023) is determined with the help of the imaging from the DESI Legacy Imaging Surveys (Zou et al. (2017); Dey et al. (2019); Schlegel et al. in prep.).The preliminary selection of targets was done in 2020 for the Milky Way Survey (MWS; Allende Prieto et al. 2020), Bright Galaxy Survey (BGS; Ruiz-Macias et al. 2020), LRG (Zhou et al. 2020), ELG (Raichoor et al. 2020), QSO (Yèche et al. 2020).
DESI started its first light observation in 2020 and will make public its Early Data Release (EDR; DESI Collaboration et al. 2023a) and the Siena Galaxy Atlas (SGA; Moustakas et al. in prep.) in 2023.EDR contains LSS catalogues that include redshift measurements, their corresponding random catalogues, and the clustering output (DESI Collaboration et al. (2023b); Lasker et al. in prep.).The One Percent Survey is a part of the EDR.It has covered around one per cent of the 5-year sky footprint and observed more than 90,000 LRGs, 270,000 ELGs, 30,000 QSOs, and 150,000 low-redshift galaxies 1 http://www.sdss.org/(Bright Galaxy Sample, BGS) (DESI Collaboration et al. 2023a).Despite the smaller numbers of galaxies and QSOs compared to the SDSS data, the number densities of tracers from the DESI One Percent Survey are larger than those of BOSS and eBOSS (introduced later in Table 2).Additionally, the rate between the observed targets and the targets of the One-Percent Surveys is larger than 85 per cent for all tracers (DESI Collaboration et al. (2023b);Lasker et al. in prep.).Thus, data from the One Percent Survey are sufficient for small-scale clustering analysis, such as the galaxy-halo connection study.
The relationship between haloes and galaxies is crucial for the modelling of galaxy clustering.However, this relation is highly nonlinear and generally subject to local environmental effects (e.g., Tinker et al. 2011;Wetzel et al. 2012).SubHalo Abundance Matching (SHAM, Kravtsov et al. 2004;Tasitsiomi et al. 2004;Conroy et al. 2006;Behroozi et al. 2010) is an intuitive empirical method to model this non-linear relation based on -body simulations that resolve hierarchical structures, including both haloes and subhaloes.This method assigns the most massive or brightest galaxy to centres of the most massive haloes in the case of central galaxies and subhaloes for satellite galaxies.The resulting probability of hosting a central/satellite galaxy in a halo/subhalo is a function of their (sub)halo mass, ( halo ).The shape of this probability is related to the stellar properties determined empirically.
As clustering observations and simulations improve, more advanced versions of SHAM algorithms are developed.For instance, Tasitsiomi et al. (2004) introduced a Gaussian scattering with dispersion  to the halo mass, to model the Gaussian residual in the galaxy-halo mass relation (e.g., Willick et al. 1997;Steinmetz & Navarro 1999).Trujillo-Gomez et al. (2011) and Reddick et al. (2013) proposed using the peak maximum circular velocity,  peak , instead of the halo mass,  vir , as it is closely associated with stellar mass and it is immune to the tidal stripping of subhaloes and pseudo evolution of their  vir .Favole et al. (2016); Rodríguez-Torres et al. (2017) proposed a SHAM implementation that takes into account the fact that ELGs and QSOs are incomplete in the massive stellarmass end.There are also SHAM variants that make use of secondary halo/galaxy properties (e.g., Hearin et al. 2013;Favole et al. 2022).SHAM methods can also include assembly bias and orphan galaxies (Lehmann et al. 2017;Behroozi et al. 2019;Contreras et al. 2021b;DeRose et al. 2022).
In this work, we use two SHAM implementations that are essentially variants of one algorithm: {,  ceil ,  smear }-SHAM ( smear -SHAM hereafter) and {,  ceil ,  sat }-SHAM (  sat -SHAM hereafter).The  smear -SHAM was used to study BOSS/eBOSS LRGs (Yu et al. 2022, Yu22 hereafter) and here we use it to model LRGs and QSOs.Here, we introduce the  sat -SHAM to be able to correctly reproduce the clustering of DESI ELGs.The free parameters in these SHAMs model the following aspects: the scatter in the galaxy-halo mass relation, ; an upper limit of -scattered  peak set by  ceil (in percentage), which reduces the possibility of massive (sub)haloes hosting a given type of galaxy or QSO; the uncertainty in spectroscopic redshift determination,  smear ; and the fraction of satellite galaxies,  sat , which we find to be only needed for reproducing the clustering of ELGs.
This paper is arranged as follows.In Section 2 we describe the early data release of DESI, including repeat observations and statistical redshift uncertainty measurements, the UNIT -body simulation, and the covariance matrix.The SHAM implementation and fitting are introduced in Section 3. In Section 4, we present the best-fitting results of SHAM and the interpretations of parameters for different tracers.We conclude our findings in Section 5.
This paper is one of the first series papers from the DESI galaxyhalo connection topical group.Papers released with EDR for One Percent Survey analysis that utilise AbacusSummit (Maksimova et al. 2021) simulations are Yuan et al. (2023) for LRG and QSO HOD, Rocher et al. (2023) for ELG HOD.Prada et al. (2023) is an overview for SHAM based on Uchuu (Ishiyama et al. 2021).A stellar-mass-split abundance matching applied on CosmicGrowth (Jing 2019) is also used to study DESI LRG-ELG cross-correlations (Gao et al. 2023).Other works will be published along with later data releases.

DESI Early Data Release
DESI, a 5-year spectroscopic survey, started instrumental tests in 2020 to ensure the 5000 fibres controlled by the robotic positioners could work properly in the focal plane over a 3-degree field of view (DESI Collaboration et al. 2022;Silber et al. 2022;Miller et al. 2023).After the commissioning, DESI conducted its Survey Validation campaign (DESI Collaboration et al. 2023a).It aims to validate the spectra reduction pipeline (Guy et al. 2023), assess the quality of data provided by Redrock 2 that derives the target type and redshift from spectra (Bailey et al. in prep.), and optimise the target selection (Schlafly et al. 2023) and fibre assignment (Raichoor et al. in prep.) of DESI.During the campaign, it explores target selection criteria broader than those of the 5-year main survey and observes objects typically four times longer than the main survey; in addition, to perform the visual inspection, few tiles are observed approximately 10 times longer than in the main survey (Alexander et al. 2023;Lan et al. 2023).For this reason, there are many repeat observations for each object.Later in April and May of 2021, DESI observed its One Percent Survey that covered about 1 per cent of the footprint of the 5-year main survey and used target selection criteria close to those of the main survey (Cooper et al. 2022;Hahn et al. 2022;Zhou et al. 2023;Raichoor et al. 2023;Chaussidon et al. 2022).The observation field is composed of 20 non-overlapping rosettes, each observed at least 12 times.This ensures very high fibre assignment completeness (larger than 85 per cent for ELGs and over 94 per cent for the rest of the tracers) in this region (DESI Collaboration et al. (2023a,b);Lasker et al. in prep.).As there are more exposures for objects that do not have a reliable redshift after the first observation, data from the One Percent Survey have a high redshift-success rate.

Galaxy Weights
To obtain an unbiased measurement in the galaxy clustering, we employ the FKP weight  FKP , the pairwise-inverse-probability (PIP) weight (Bianchi & Percival 2017), and the angular-up weight (ANG) (Percival & Bianchi 2017) for pairs of galaxies.In the calculation of the effective redshift, we use the total weight  tot  tot =  FKP  comp . ( where  comp is the fibre-assignment completeness weight provided in the LSS catalogue.We briefly describe all of them below and we refer the readers to DESI  (Feldman et al. 1994) that minimises the variance in the clustering estimator (see Section 3.1) when the observed number density of tracers varies with redshift where () is the average number density at redshift , and  0 is the amplitude of the observed power spectrum at  ≈ 0.15 ℎ Mpc −1 . 0 = 10000, 4000, 6000 ℎ −3 Mpc 3 for LRGs, ELGs and QSOs respectively (DESI Collaboration et al. 2023b).
The PIP+ANG weighting scheme has been developed to correct the missing galaxy pairs due to fibre collision.Mohammad et al. (2020) have proved that PIP+ANG weights provide an unbiased clustering down to 0.1 ℎ −1 Mpc.So the clustering measurement provided by the EDR has implemented this weighting scheme in addition to  FKP (Section 3.1).The  comp provided in the LSS catalogues is for correcting the observational incompleteness due to the fibre assignment (DESI Collaboration et al. 2023b).PIP and ANG weights, as well as  comp , are all calculated with simulations of fibre assignment as described in Lasker et al. in prep..We calculate the effective redshift of pairs of galaxies at redshifts   and   (e.g., Bautista et al. 2021), with The effective volume  eff,obs (Wang et al. 2013) also involves  0 as where Δ (  ) is the comoving survey volume and (  ) is the mean number density of the tracer inside the redshift bin   .The effective number density is calculated as We present in w v = 30.3+0.2 0.2 v = 96.9+1.9 1.9 1000 0 1000 v(km s 1 ) 10 3 10 2 10 1 10 0 2.1 < z < 3.5, outliers: 0.6% w v = 88.8 +0.8 0.8 v = 291.3+3.9 3.9 The first columns of all rows are results for total samples, while the rest are for sub-samples at redshift bins.The statistical redshift uncertainty measured by Lorentzian functions  Δ (solid red lines) and standard deviations σΔ of Δ is presented in the label of each subplot.For LRG samples, we also fit Δ histograms with Gaussian functions (blue solid lines), providing their best-fitting dispersion  Δ in the label as well.The fraction of Δ that are not included in the fittings is indicated as outlier fractions in titles.

Repeat Observations and Statistical Redshift Uncertainty
The spectroscopic measurements of redshift have associated uncertainties (i.e., the redshift uncertainty) due to factors like the spectral line width, observing conditions, and different astrophysical effects.
The impact of the redshift uncertainty is equivalent to adding stochasticity to the peculiar velocity of the observed object, and thus will bias the measurement of anisotropic clustering and velocity bias (Guo et al. 2015;Hou et al. 2018;Yu et al. 2022).The redshift uncertainty can be quantified by repeat observations statistically and via its influence on the clustering using our SHAM method (see Section 3.2).The results of those two estimators should be consistent.
Objects observed repetitively exist in all stages of Survey Validation.However, the ones from the One Percent Survey are biased towards faint objects by design (see Section 2.1).Therefore, we used data from the early stage of Survey Validation to obtain an unbiased estimate of the redshift uncertainty.The redshift difference, Δ, is calculated for all pairs of repeated spectra for each object and then converted to radial velocity using Δ = Δ/(1 + ), where  is the speed of light and  is the mean redshift of the pairs.Δ measurements larger than the redshift failure threshold (1000 km s −1 for LRGs and ELGs, 3000 km s −1 for QSOs) are then removed.
The histograms of the redshift difference of ELGs, LRGs, and QSOs are presented in Fig. 1 in black dots.The error bars of those histograms are calculated using the delete-one jackknife method.Our fitting range is around [−200, 200] km s −1 for LRGs, [−150, 150] km s −1 for ELGs, and [−1600, 1600] km s −1 for QSOs except for [−500, 500] km s −1 for QSOs at 0.8 <  < 1.1.The title of each subplot in Fig. 1 shows the percentage of Δ measurements that are beyond the fitting range.For SDSS BOSS/eBOSS LRGs, the redshift difference in all redshift ranges can be well fitted by Gaussian functions (Ross et al. 2020;Lyke et al. 2020;Yu et al. 2022).For DESI, all tracers show a preference for Lorentzian distributions in general (solid red lines in Fig. 1) as In Eq. ( 6),  is a normalization factor,  is the location of the peak value on the x-axis, and 2 Δ is the full-width-half-maximum of the Lorentzian distribution.In addition, we also try to describe Δ histograms of LRGs with Gaussian profiles N (,  Δ ) (solid blue lines in Fig. 1).We will discuss which profile to use for SHAM in Section 4.4.As  and  are well consistent with 0, we only present the best fitting Lorentzian  Δ and Gaussian  Δ on the labels of Fig. 1.We also calculate the standard deviation of the redshift difference σΔ .
In Fig. 1, we observe a much smaller redshift uncertainty for the ELGs than that of the LRGs and QSOs.The maximum  Δ of ELGs is 13.4 ± 0.1 km s −1 , while the minimum  Δ of LRG and QSOs is 34.6 ± 0.9 and 30.2 ± 0.2 km s −1 respectively.This can be attributed to the narrow [O ii] emission for ELG redshift determination, compared with absorption lines of LRGs and broad emissions of QSOs.Additionally, galaxy samples (LRGs and ELGs) show increasing uncertainty with redshift.This is because galaxies are fainter at higher redshift, and spectral lines for redshift determination have a decreasing signal-to-noise ratio and larger uncertainty.But for QSO this is not the case, as QSOs at higher redshifts are not necessarily fainter.Another reason for the non-monotonic QSO  Δ trend is that the measurement made by repeat observation is no longer reliable at  ≳ 1.5.We will explain this in detail in Section 4.4.
We use UNIT halo catalogues with subhaloes identified by the rockstar (Behroozi et al. 2013a) halo finder that provides properties at the current snapshot, such as positions, peculiar velocities, virial mass  vir , and the maximum circular velocity  max .We regard  vir of haloes with more than 50 dark matter particles to be reliable, i.e.,  good vir > 6 × 10 10 ℎ −1 M ⊙ .The merger/stripping histories of haloes and subhaloes are provided by consistent trees (Behroozi et al. 2013b).They are used to determine their peak maximum circular velocity throughout the accretion history, i.e.,  peak , which is the proxy of halo mass in our SHAM study.
UNIT simulations are created using the fixed-amplitude method implemented in pairs of simulation boxes to suppress the cosmic variance (Angulo & Pontzen 2016;Chuang et al. 2019).The effective volume of UNIT simulations is much larger than those of DESI EDR tracers as shown in Table 2.So we can take just one simulation box in each snapshot and ignore the influence of the UNIT cosmic variance on our SHAM fitting.
UNIT includes 128 snapshots of simulations from redshift 99 to 0 and we employ 14 of them with their redshift presented in the fourth column of Table 2.We select the UNIT snapshot whose redshift is the closest to the  eff (Eq.( 3)) of the corresponding DESI sample among all the snapshots.

Galaxy Clustering
The two-point correlation function (2PCF) measures the excess probability of finding a galaxy pair compared to a random distribution in a given volume.For observations, we use the Landy-Szalay estimator (LS; Landy & Szalay 1993) which minimises the variances of the measurements for an irregular geometry: 3 http://www.unitsims.org/ where the data-data (DD), data-random (DR), and random-random (RR) pair counts are normalized by their corresponding total number of pairs. and the pair counts can be calculated as a function of the pair separation  and  which is the cosine of the angle between the line connecting the galaxy pairs and the line-of-sight. FKP is applied to every individual galaxy in the data and random catalogue.PIP weights are applied to DD pair counts, and ANG weights are implemented to both the DD and DR pairs.The SHAM galaxies are populated in periodic boxes based on halo catalogues from the UNIT -body simulation, so we use the Peebles-Hauser estimator (PH;Peebles & Hauser 1974) to obtain their 2PCF as follows: Unlike observation that requires random catalogues to calculate RR pairs, we use the following expression to calculate them analytically in the simulation box: where  max and  min are the boundaries of the separation bins,  box = 1 ℎ −3 Gpc 3 is the volume of the UNIT simulation box, and   = 200 is the number of  bins.By weighting the 2D  (, ) with Legendre polynomials  ℓ (), we obtain the 1D  multipoles as We fit our SHAM to observations based on the monopole and quadrupole, i.e., ℓ = 0, 2. We use 10 logarithmic  bins in (5, 30) ℎ −1 Mpc and 200  bins in (−1, 1).The projected 2PCF is calculated for cross-checking the clustering of the best-fitting SHAM galaxies.This is calculated as where  max = 30 ℎ −1 Mpc to avoid the contamination of the systematics on larger scales as shown in Yu22.pycorr and Corrfunc Python packages (Sinha & Garrison 2019;Sinha & Garrison 2020) are used to calculate  ℓ () and   .
In observations, 2PCFs are calculated in redshift space.So the position of our mock galaxies produced by SHAM should take into account the redshift-space distortion (RSD; Kaiser 1987) using: where  is the coordinate in the -axis which is the line of sight, and its subscripts 'redshift' and 'real' illustrate that the coordinate is in the redshift space or in the real space. pec,Z is the proper peculiar velocity of SHAM galaxies along the -axis, and  is the redshift of the simulation snapshot.As the cosmic variance of UNIT simulations is small, we can safely ignore the variations in quadrupoles for different line-of-sights (Smith et al. 2021).

SHAM Implementation
SubHalo Abundance Matching (SHAM) is an empirical method to construct a realistic, monotonic galaxy-halo mass relation based on -body simulations.In its simplest form, a SHAM has a single free parameter  relating the masses of galaxies and haloes and can successfully reproduce the observed clustering (e.g., Tasitsiomi et al. 2004;Behroozi et al. 2010).As observations provide the clustering of multiple tracers with higher and higher accuracy, this prototype should also be improved.We thus introduce the massive (sub)halo incompleteness  ceil , the redshift uncertainty  smear , and a free satellite fraction  sat in the SHAM implementation besides the galaxy-halo mass scatter .Their impact on the 2PCF  ℓ () and projected 2PCF   (  ) are presented in Appendix A.
In our study, all (sub)haloes in the simulation have their  peak multiplied by an asymmetric Gaussian as to avoid negative  scat .
Then those (sub)haloes are sorted in descending order of  scat and the first  ceil  UNIT /100 ones are removed. UNIT is the total number of haloes and subhaloes in this UNIT simulation.It means the most massive (sub)haloes will not be assigned with a galaxy/QSO in its centre. ceil is introduced for target selections that possibly remove some of the most massive LRGs, resulting in incompleteness in the host (sub)halo mass.ELGs are mostly star-forming galaxies and thus are not expected to be complete in stellar mass and thus (sub)halo mass (e.g Gonzalez-Perez et al. 2020;Hadzhiyska et al. 2021).This is because the hot and dense centre of massive (sub)haloes is an environment that depletes the cold gas and thus stops star formation (e.g.Kauffmann et al. 2004;Dekel & Birnboim 2006;Peng et al. 2010).QSOs are bright active galactic nuclei (AGN), i.e., their supermassive black holes actively accrete cold gas via discs (e.g., Rosario et al. 2013).In the semi-analytical models (SAM), the formation of AGNs with  bol ≳ 10 45.1 erg s −1 only happens at haloes with  vir ≲ 10 13 ℎ −1 M ⊙ during starbursts (Griffin et al. 2019).Uchiyama et al. (2018) attribute the absence of QSOs in the overdense regions (i.e., massive haloes) at  ∼ 2-3 to the lack of wet mergers which leads to the QSO activity.In hydrodynamical simulations, Weinberger et al. (2018) also find that AGNs exit their high-accretion phase (i.e., the QSO phase) in the most massive galaxy at  ∼ 2. The absence of QSOs in those massive quenched galaxies means their absence in the most massive haloes.So LRGs, ELGs and QSOs all require the  ceil truncation, which still allows (sub)haloes with large  peak with the help of .We need to point out that the actual format of the massive halo incompleteness should not depend solely on  peak .This  ceil truncation is chosen as it is the simplest implementation for the incompleteness and it enables a good description of the observed 2-point clustering (See Section 4.1). smear -SHAM and  sat -SHAM algorithms then deviate after this step.
For  smear -SHAM, we populate a central/satellite galaxy in the centre of each halo/subhalo in the  ceil -truncated catalogue from the most massive ones to the least ones until we get the expected number of SHAM galaxies where  box = 1 ℎ −3 Gpc 3 is the box size of the UNIT -body simulation. eff is the effective number density of the observed sample obtained using Eq. ( 5) and the values for each galaxy sample are presented in Table 2.The proper peculiar velocity of the host (sub)haloes  ℎ pec is also assigned to their galaxies.The velocity of the galaxy along the line of sight   pec,Z is then blurred by  smear to mimic the effect of the redshift uncertainty as where  ℎ pec,Z is the component of  ℎ pec on the -axis, N (0,  smear,G ) and L (0,  smear,L ) are a random number sampled by a Gaussian profile or a Lorentzian profile, respectively (as discussed in Section 2.2).As the standard Lorentzian profile is heavy-tailed and subexponential, we remove L (0,  smear ) larger than 400 km s −1 for LRGs and 2000 km s −1 for QSOs.We do not use the  smear parameter in  sat -SHAM as explained in Section 4.4.
In  sat -SHAM, we further separate haloes and subhaloes from the  ceil -truncated catalogue.Only the first  sat  gal /100 subhaloes are kept as hosts of ELG satellites and the first (1 −  sat ) gal /100 haloes are for central ELGs.Then we assign the centre position and the proper peculiar velocity of those selected halo/subhalo to their central/satellite galaxies.
Note that in  smear -SHAM, the satellite fraction  sat is defined as the percentage of subhaloes in the list of (sub)haloes selected by SHAM, that is, where  sat is the number of satellite galaxies in the SHAM galaxy catalogue,  sub,SHAM =  sat is the number of subhaloes selected by SHAM, and  halo,SHAM is the number of haloes selected by SHAM.
Note that  sat is different from the percentage of subhaloes in the UNIT simulations: where  sub is the total number of subhaloes in the UNIT simulation and  halo is the total number of haloes there.Finally, we calculate the clustering of model galaxies in redshift space produced by  smear -SHAM (LRGs, QSOs) or  sat -SHAM (ELGs) and compare it with observations, trying to find the bestfitting parameters.As shown in Appendix A, ,  ceil ,  smear and  sat are the primary factors that affect the spatial distribution of DESI dark matter tracers at 5-30 ℎ −1 Mpc.Given the well-reproduced clustering (see Section 4.1), we do not explore additional effects such as the assembly bias, which can not be well constrained by our DESI sample due to its low number density (Contreras et al. 2021a;Yuan et al. 2023;Rocher et al. 2023).However, to describe the galaxy-halo connection of dense tracers like the Bright Galaxy Sample (BGS; Pearl et al. 2023), and the cross-correlation between different tracers (Gao et al. (2023), Yuan et al. in prep), assembly bias will play a role.In addition to the 3-parameter SHAM, we further discuss the performance of the complete 4-parameter SHAM {,  ceil ,  smear ,  sat } in Appendix A.

SHAM Constraints
We try to find the best-fitting SHAM parameters using a Monte-Carlo sampler Multinest4 (Feroz & Hobson 2008;Feroz et al. 2009Feroz et al. , 2019) ) assuming a Gaussian likelihood L (Θ) for our parameter constraint The  2 values are obtained as where Θ = {,  ceil ,  smear } for LRG and QSO samples and Θ = {,  ceil ,  sat } for ELG samples. = ( 0 ,  2 ) denotes the vector composed of the 2PCF monopole and quadrupole.The subscripts 'data' and 'model' of  represent measurements from the observational data and SHAM mocks, respectively.C is the unbiased covariance matrix that should include the variances of  data and  model .
The variances of  model can be further decomposed into the cosmic variance of UNIT simulation and the statistical variance due to the random processes included (Section 3.2).The variance of UNIT is considered to be negligible (Section 2.3). model is obtained by averaging the 2PCFs of 32 SHAM galaxy realizations generated using the same Θ with different random seeds.Because the statistical variance of 32 realizations is less than 5 per cent of the observational errors in general, increasing the number would increase the computing cost without much gain in the reliability of the parameter constraint.So C can be estimated as the variance of  data via (Hartlap et al. 2007): where  bins = 20 (Section 3.1) is the length of  data , i.e., the total number of bins of the monopole and the quadrupole used in the SHAM fitting.C s is the jackknife covariance matrix, and is calculated using pycorr5 with  mock = 128 jackknife subsamples of the observational data.C s is thus expressed as where  ( ) is the correlation function measured from the data with the  th jackknife subsample removed, and is the mean 2PCF of all jackknife subsamples.The errors for the data vector are the square root of the diagonal terms of C.
We employ Multinest, an efficient nested sampling technique, to constrain Θ.We keep the default convergence criteria which is a tolerance of 0.5 and use 200 particles for the sampling.Using a smaller tolerance or more particles takes more computing time but provides a similar posterior.The prior range for the SHAM fitting is listed in Table 1.The best-fitting parameters, which are the medians of the 16th and 84th percentiles (the 1- confidence limit) of the marginalized posterior distributions of individual parameters (Appendix B), their 1- confidence limits, and the minimum  2 are provided by PyMultinest6 (Buchner et al. 2014).All the derived quantities, i.e.,  sat (in  smear -SHAM), halo occupation distribution (HOD), the probability of a (sub)halo to host a central (satellite) galaxy (PDF), the mean halo mass ⟨ vir ⟩ and the mean  peak ⟨ peak ⟩ are computed using the nested sampling chain.compared with that of the best-fitting SHAM model galaxies with its statistical uncertainty (solid lines with shades).Monopoles and their residuals normalised by the observed errors  obs are presented in the first and the second rows.The third and fourth rows present those for quadrupoles.Each colour shows a different redshift range, as indicated in the legend.The first column shows results for LRGs in different redshift bins and the second for the total sample.The error bars of data are obtained from 128 jackknife samples, and the statistical uncertainty of SHAM galaxies indicated in the width of the shades is the standard deviation of its 32 realizations divided by √ 32.The uncertainty of best-fitting LRG SHAM galaxies is too small to be seen.Our SHAM provides good fit to the observed clustering at 5-30 ℎ −1 Mpc.

RESULTS
We present results of  smear -SHAM for LRGs and QSOs,  sat -SHAM for ELGs for the DESI One Percent Survey in this section.The bestfitting 2PCF, features of the best-fitting ,  ceil ,  smear and  sat for different tracers are discussed respectively.We also check the consistency between the HOD measured from our best-fitting SHAM with those from HOD studies using the same data.

Clustering
We fit the 2PCF multipoles of the LRG, ELG and QSO samples from the DESI One Percent Survey at scales of 5-30 ℎ −1 Mpc over the redshift range 0.4 <  < 3.5 with our SHAM algorithms.Table 2 summarises the best-fitting parameters and their corresponding 1- confidence intervals, as well as the minimum  2 divided by the number of degrees of freedom.Note that  smear -SHAM results presented in Table 2, Figures 2-4 and in the appendices all use a Lorentzian  smear,L .In this case,  sat is obtained as a derived parameter from the Monte-Carlo chain.
Figures 2-4 are the 2PCF monopole (first row) and quadrupole (third row) of the observed tracers (filled circles with error bars) and model galaxies/QSOs generated using SHAM with the minimum- 2 parameter set (solid lines).The shaded area around the SHAM clustering is the standard deviation of 2PCFs for all 32 SHAM re- The columns are: 1) observed tracer type, 2) redshift range, 3) effective redshift calculated using Eq. ( 3), 4) redshift of the UNIT simulation snapshot for the SHAM fitting that is close to  eff , 5) the effective volume  eff of the observed tracer at the corresponding redshift range obtained with Eq. ( 4), 6) the effective number density  eff calculated with Eq. ( 5) multiplied by 10 4 , the best-fitting parameters, i.e., 7) , 8) the redshift uncertainty  smear,L , 9) the massive-(sub)halo incompleteness  ceil , 10) the satellite fraction  sat and 11) the minimum  2 divided by the degree of freedom. smear of ELG samples are asserted to 0 and the satellite fraction of LRGs and QSOs is a derived parameter from the nested sampling chain.alizations divided by √ 32.The observed error-rescaled residuals are presented in the second (monopole) and fourth (quadrupole) rows.
The observed clustering of LRG samples is well-fitted by  smear -SHAM as shown in Fig. 2. The reduced  2 value of LRG fitting at 0.8 <  < 1.1 is around 1.7, which could be explained by underfitting.However, our SHAM LRGs at 0.8 <  < 1.1 also reproduce   at 5 <   < 30 ℎ −1 Mpc (see Appendix C for the consistent projected 2PCF of SHAM LRGs and observations at this redshift bin).We attribute this large value to the off-diagonal terms in its jackknife covariance matrix.SHAM LRGs at 0.4 <  < 0.8 have an underestimated  2 at  > 20 ℎ −1 Mpc.At these scales, observations present a plateau while the quadrupoles of SHAM LRGs decrease.This flat quadrupole pattern is also present at 20 − 40 ℎ −1 Mpc for LRGs from both SDSS-BOSS SGC at  > 0.4 and eBOSS LRGs at all redshift bins, even after eliminating all known observational systematics (Ross et al. 2017;Zhao et al. 2021).The observed quadrupole resumes the smooth trend indicated by models at  > 50 ℎ −1 Mpc.Thus, the observed plateau could be attributed to cosmic variance or some uncorrected systematics for LRGs at  > 0.4.The SHAM 2022), in particular for red galaxies (Ross et al. 2014).A detailed investigation of this problem is left for future work.ELG multipoles are well reproduced by SHAM galaxies at all redshift ranges as shown in Fig. 3 and indicated by the reduced  2 values in Table 2. QSO clustering has large observed errors due to the small number density of QSOs, which leads to large shot noise.Fig. 4 proves that  smear -SHAM provides a consistently good description of the observation of QSOs in a large redshift range from  = 0.8 to  = 3.5.
With the best-fitting catalogues of SHAM galaxies/quasars, we calculate their power spectrum with pypowspec7 for the linear bias

𝑏 lin via
where  g (, ) is the power spectrum of SHAM galaxies at redshift ,  lin m (,  = 0) is the linear matter power spectrum used by UNIT simulations renormalized to  = 0.Both power spectra are in real space. () is the linear growth rate at redshift . < 0.05 ℎ Mpc −1 in Eq. ( 23) means that the result is obtained by averaging over this  range.17) is plotted in dashed lines.
redshift compared to the case at high redshift, resulting in an increasing bias with respect to the redshift.Studies using the same DESI EDR data show similar trends and consistent values for  lin (Yuan et al. 2023;Rocher et al. 2023;Prada et al. 2023).

Scatter 𝜎 in galaxy-halo mass relation
As discussed in Yu22,  in our  smear -SHAM is composed of the intrinsic scatter in the galaxy-halo mass relation and the completeness for galaxies with an intermediate stellar mass.For LRG samples,  ∼ 0.2 dex.However, since there is a degeneracy between  and  ceil (see Appendix B for the posteriors of LRGs), it is not clear whether there is a redshift evolution in .For ELG samples, the constraints in  are weak regardless of the prior range.Given its large number density, this is not the result of large errors in clustering, as is the case for QSOs.The (sub)halo incompleteness of ELGs is related to their incompleteness in stellar mass and in luminosity (Favole et al. 2016;Gonzalez-Perez et al. 2020).This leads to a complex galaxy property-halo mass connection for ELGs, thus a weakly constrained  as it integrates many factors. also leads to the stochastic variance in the clustering of SHAM galaxies.For the LRG and ELG samples, this variance is as small as 5 per cent of the observational error  obs (not visible as shown in Fig. 2-3).So we can also ignore its effect on the final  2 values in general.

Massive (Sub)Halo Incompleteness 𝑉 ceil
For LRGs,  ceil describes the halo/stellar mass incompleteness at the massive end.As shown in LRG posteriors (See Appendix B),  ceil values for LRGs are very small but are definitely non-zero at 1- level.This incompleteness can be attributed to the fact that the target selection of LRGs removes some of the most massive blue galaxies, leading to empty massive haloes.
For ELG samples,  ceil is critical to describe their absence in the centre of massive (sub)haloes.ELG entries in Table 2 show that we shall remove a per-cent level of (sub)haloes, allowing few (sub)haloes with  peak > 300 km s −1 to host an ELG at their centre (Fig. 13).Since there is no degeneracy among parameters of the  sat -SHAM, the 1 −  difference between the  ceil of ELGs at 0.8 <  < 1.1 and 1.1 <  < 1.6 embodies their clustering difference.
QSOs at 0.8 <  < 1.1 show a larger  ceil than that of QSOs at higher redshifts, but this difference is not significant due to the weak constraint.This is consistent with Chaussidon et al. (2022) in which QSOs at  < 1.0 show a smaller purity than those at high redshifts.Even though  ceil for QSOs only exclude less than 1 per cent of (sub)haloes with the largest  scat (Eq.( 13)), the clustering of QSOs cannot be well-fitted without the  ceil parameter.As explained in Section 3.2, this is consistent with findings of SAM studies (e.g., Griffin et al. 2019) and observations (e.g., Uchiyama et al. 2018), in which QSOs are absent in the centre of the most massive (sub)haloes.

Redshift Uncertainty 𝑣 smear
smear -SHAM uses a Lorentzian  smear,L by default as this is a good model for DESI tracers in general.In particular, for LRGs, Lorentzian and Gaussian profiles can both describe the redshift difference distribution of LRG repeat observations (Fig. 1).So we perform  smear - SHAM fitting with Gaussian  smear,G as well and Table 3 includes the best-fitting results.The  2 values of SHAM with  smear,G are similar to those of SHAM with  smear,L .Their best-fitting  and  ceil are also consistent with each other, meaning that the clustering effect of a truncated Lorentzian  smear,L and a Gaussian  smear,G is almost equivalent.So we need to check the  smear consistency with the statistical redshift uncertainty estimated from repeat observations.Fig. 6 is the comparison between the best-fitting SHAM  smear (filled stars with error bars) and the best-fitting redshift uncertainty measured from repeat observations (empty squares with error bars) for DESI LRGs (left panel) and QSOs (right panel).The results of Lorentzian profiles are in red and those of Gaussian profiles are in blue.In particular, Gaussian profiles for LRGs are vertically shifted.The best-fitting LRG SHAM  smear,G values agree with the Gaussian dispersion  Δ of repeat observations, while SHAM  smear,L values underestimate the statistical redshift uncertainty evaluated using the width of Lorentzian functions  Δ .Note that the standard deviation σΔ of LRG repeats are also consistent with the dispersion of the Gaussian profile.Therefore, the Gaussian profile  smear,G is more suitable for illustrating the uncertainty of the redshift of LRGs in redshift bins.
In contrast, the Lorentzian profile results ( Δ = 46.5±0.8 km s −1 while  smear,L = 40 ± 9 km s −1 ) are in better agreement for LRGs in the full redshift range, indicating that there are multiple types of LRGs with different redshift uncertainty properties.As a result, Raichoor et al. (2023); Chaussidon et al. (2022) use a linear combination of Gaussian profiles to fit the redshift difference Δ.Nevertheless, a truncated Lorentzian with just one parameter works in the same way as multiple-Gaussian profiles in terms of the clustering effects and SHAM results.
For QSOs in the right panel of Fig. 6, SHAM  smear,L shows a non-decreasing trend.This reflects the quadrupole amplitude of QSOs at different redshift bins shown in the third row of Fig. 4.However, this trend is inconsistent with that of repeat observations (red empty error bars), and SHAM  smear,L starts to deviate from  Δ of repeat observations at  ≳ 1.5.The discrepancy is possibly due to the switch of the main spectral line for redshift determination from Mg ii and C iv (Zarrouk et al. 2018).QSO spectral lines are subject to systematical velocity shifts caused by astrophysical effects (e.g., Gaskell 1982;Richards et al. 2002Richards et al. , 2011) ) and repeat observations for the same object cannot capture this shift.But this shift is different from object to object, creating an extra relative random motion between QSO pairs.Moreover, Shen et al. (2016) have proved that Mg ii is the least shifted broad emission of QSOs, while C iv can be strongly shifted.This will result in a larger random motion between QSO pairs at  > 1.5 than those at  < 1.5.This motion is integrated into  smear,L of SHAM, resulting in its fast rise at  > 1.5 and explaining the inconsistency between  smear,L and  Δ .
The redshift uncertainty of ELGs is small in general as presented in Fig. 1.Its largest redshift uncertainty among ELGs samples, i.e.,  smear,L = 13.4 km s −1 for ELGs at 1.1 <  < 1.6, does not produce a significant clustering effect, as illustrated in Fig. 7.The magenta line is the clustering of SHAM galaxies created using the best-fitting  sat -SHAM.The blue line shows the clustering with  smear,L = 13.4 km s −1 applied to the peculiar velocity of the SHAM ELGs.For the monopole,  smear,L have little influence as expected, and the influence on the quadrupole is restricted to 5-8 ℎ −1 Mpc and is within 1- range of the observed jackknife error bars.Therefore, asserting  smear = 0 for  sat -SHAM does not bias the best-fitting results of the other three parameters.

Satellite Fraction 𝑓 sat
The fraction of satellites,  sat , for LRGs and QSOs, is calculated when  and  ceil are given to  smear -SHAM.We present in Fig. 8 the impact of ,  ceil and the number density  eff on  sat for SHAM galaxies produced by  smear -SHAM at  = 0.94 (blue) and  = 1.83 (red).We  sub of UNIT simulations at those redshifts are plotted in dashed horizontal lines, obtained using Eq. ( 17).The fixed parameters8 are  = 0.1,  ceil = 0.1,  eff = 4 × 10 −4 Mpc −3 ℎ 3 . sat monotonically increases with ,  ceil and  eff .This is because larger ,  ceil and  eff all mean selecting more (sub)haloes with small  peak , corresponding to a larger fraction of subhaloes/satellites.In contrast, for ,  ceil ∼ 0, there are a few selected subhaloes that have a large  peak , resulting in  sat <  sub .Due to the tight constraints on LRG  and  ceil , the 1- confidence interval of LRG  sat , which is a derived parameter, is also small.From another aspect, the slope of the  sat - and  sat - ceil relations decreases as  and  ceil become larger.Additionally, the slope of the  sat - eff relation increases with  eff .The combination of these effects results in the small errors of QSO  sat despite its loose constraints on  and  ceil .
The satellite fraction of  smear -SHAM is also affected by the redshift, i.e., the substructure growth. sat of SHAM galaxies at  = 1.83 are lower than those at  = 0.94, calculated with the same  smear -SHAM parameters.This is consistent with the decreasing trend of  sat for LRGs and QSOs with the redshift.
For ELGs, we find that about 4 per cent of them are satellite galaxies when we fit the data with our  sat -SHAM method.Such a low fraction of satellites is also found in models mimicking a DESI-like survey (Gonzalez-Perez et al. 2018).In the literature, for different selections of ELGs, this fraction has been found to range from  sat ∼ 5 to ∼ 22.5 per cent (Favole et al. 2016;Gao et al. 2022) in VIPERS and from ∼ 17 per cent (Guo et al. 2019) to 19.3 per cent (Lin et al. 2023) where  ∈ (0.8, 1.6), and  thres [O ii] ( = 0.5) = 10 41.75 erg s −1 , which is the  [O ii] lower bound of VIPERS ELGs with  sat = 7.0 ± 2.0 per cent.This fraction for VIPERS and eBOSS are 10 and 12 per cent respectively.Thus, it is reasonable if we obtain a smaller  sat from 0 500 1000 1500 V peak (km s 1 ) 10 3 10 2 10 1 10 0 probability LRG ELG QSO Figure 13.The probability of a (sub)halo to host an LRG (red), an ELG (blue) or a QSO (orange) as a function of  peak of (sub)haloes for LRGs with Gaussian  smear,G , ELGs and QSOs at 0.8 <  < 1.1.The shades are their 1- errors calculated using the Monte-Carlo chain.
the DESI data, compared to that of the eBOSS ELGs and VIPERS samples.
However, it is still possible that our satellite fraction is underestimated as we do not include orphan galaxies that are necessary for correcting the deficits of the current subhalo tracking method (Behroozi et al. 2019).We check in Appendix D the consistency between the  sat measured by  sat -SHAM and galaxy mocks provided by UniverseMachine (Behroozi et al. 2019) and SAM models (Gonzalez-Perez et al. 2018).The mass resolution of the UNIT simulation may also not be good enough to resolve all substructures for ELGs.Moreover,  sat is model dependent (e.g., Favole et al. 2016;Gao et al. 2022).In studies for DESI ELGs, Gao et al. (2023) present a redshift-and stellar-mass-dependent satellite fraction with reconstructed orphan galaxies.Rocher et al. (2023) find that adding conformity can lead to a smaller  sat compared to HOD without that.
So it is difficult to compare fairly the  sat value provided by different models for different galaxy surveys.We will leave those for future work.
In Fig. 10, we present the satellite fraction  sat as a function of the (parent) halo mass  vir .For LRGs and QSOs, almost all galaxies residing on small haloes selected by SHAM are satellites, and then  sat decreases to 0 as  vir increases to 8.9 × 10 13 ℎ −1 M ⊙ for LRGs and 1.8 × 10 13 ℎ −1 M ⊙ for QSOs.For ELGs, less than half of the selected small haloes host satellites, then the satellite fraction decreases to 0 as  vir > 2.2 × 10 12 ℎ −1 M ⊙ .

Halo Occupation Distribution
Fig. 11 shows the average halo occupation distribution (HOD) of SHAM LRGs with Lorentzian  smear,L at 0.4 <  < 1.1, SHAM ELGs at 0.8 <  < 1.6 and SHAM QSOs at 0.8 <  < 3.5 as a function of the halo mass  vir .They are computed with the weights from the Monte-Carlo chain. vir corresponds to the virial mass of host haloes or parent haloes of subhaloes.We opt not to compare directly our halo occupation with the other DESI EDR galaxy-halo connection results.This is because our HODs originate from different N-body simulations, varying in redshift, halo finders, and mass resolution.These differences might influence the HOD configuration.Consequently, rather than pursuing a direct comparison of HOD, our target in the following section is to conclude the common features of DESI EDR tracers measured by different galaxy-halo connection methods and the characteristics of different tracers provided by our SHAM.
For LRGs, their stellar mass is closely related to the halo mass (e.g., Leauthaud et al. 2012;Behroozi et al. 2019).Their HOD can be modelled using a 5-parameter form, with a smoothed step function for central galaxies, and a power law for satellites (e.g., Zheng et al. 2005;Zhai et al. 2017).The HOD of our central SHAM LRGs reaches ⟨⟩ = 0.75 ± 0.07 at  vir = 10 13.3 ℎ −1 M ⊙ and decreases towards the massive end as we set  ceil free.This incompleteness is consistent with the measurement of AbacusSummit HOD for DESI LRGs (Yuan et al. 2023).
There are various HOD models for central ELG.Avila et al. 2020;Rocher et al. (2023) discuss several ELG central profiles: the modified high-mass-quenched model (Alam et al. 2020), the Gaussian function, the star-forming HOD model (Avila et al. 2020), the lognormal HOD model (Rocher et al. 2023).The central HOD of our SHAM ELGs shows a preference for a star-forming HOD profile with a turning point at  vir = 10 11.7 ℎ −1 M ⊙ that reaches ⟨⟩ = 0.06 ± 0.03.ELGs residing in  vir > 10 12.5 ℎ −1 M ⊙ are also found in the study of Gao et al. (2023).
There are also multiple profiles for QSO HOD models (Smith et al. 2020;Yuan et al. 2023).Our central QSO HOD reaches the maximum value ⟨⟩ = 0.016 ± 0.001 after  vir = 10 12.4 ℎ −1 M ⊙ .Note that no tracer reaches ⟨⟩ = 1.It means that we will not find one galaxy/QSO in every halo above a certain halo mass.It is consistent with the  ceil results that LRGs from the One-Percent Survey are not complete, while ELGs and QSOs are absent from massive haloes due to physical reasons.
Note that our SHAM model galaxies/quasars present a decreasing number of centrals in the massive halo in Fig. 11 (red lines with shades).This is because we apply a simple, empirical truncation to the massive haloes via  ceil , aiming at recover the auto-correlations of the DESI EDR tracers with a minimum number of parameters.Therefore, the increasing incompleteness at the massive end for model galaxies is not necessarily physical, given the lack of assembly bias effect for example.In fact, Rocher et  creasing trend for central ELGs in DESI with four different models that include the assembly bias.Meanwhile, Yuan et al. (2023) provide a constant number of central galaxies/quasars in massive haloes and find incompleteness there for both LRGs and QSOs from DESI, with the HOD models including the assembly bias.Nevertheless, the fact that all these galaxy/quasar-halo connection models show a central galaxy occupation below unity regardless of the assembly bias suggests that LRGs, ELGs, and QSOs from the DESI One-Percent Survey are likely incomplete in their host halo masses.We will need a more sophisticated galaxy-halo relation as well as better observations and simulations to understand this incompleteness better.
The average HOD of our SHAM satellites for all tracers can be fitted by two exponential functions of  vir , i.e., The best-fitting results obtained by PyMultinest are presented in Table 4.The second exponent  at a lower mass range for all tracers is consistent and around 2. Though  turn values are different for different tracers, their slope in the massive end  is well consistent with 0.7.This is consistent with ELG HOD slopes in Rocher et al. (2023) but smaller than those from LRG HOD in Yuan et al. (2023).
The mean parent halo mass ⟨ vir ⟩, derived from the Monte-Carlo chain as introduced in Section 3.3, is shown in Table 4.Those values are consistent with those of AbacusSummit HOD using the same data from the DESI One Percent Survey (Yuan et al. 2023;Rocher et al. 2023).Fig. 12 is the evolution of the mean parent halo mass for SHAM LRGs with Gaussian  smear,G , ELGs, and QSOs.⟨ LRG vir ⟩ is not smaller than 10 13 ℎ −1 M ⊙ , and ⟨ QSO vir ⟩ values range from 10 12.2 ℎ −1 M ⊙ to 10 12.7 ℎ −1 M ⊙ .Both decrease with redshift, consistent with the redshift-evolution HOD results from Yuan et al. (2023).For ELG parent halo, there is no significant evolution and both ⟨ vir ⟩ are below 10 12 ℎ −1 M ⊙ .Rocher et al. (2023) and Gao et al. (2023) also find the same feature for the mean halo mass of ELGs but with slightly different values.Those are all consistent with the expectation of DESI Collaboration et al. (2016a).But the bias of ELGs still increases with redshift as indicated in Fig. 5.It should be noted that only 1.6 per cent of ELGs and less than 0.03 per cent of LRGs and QSOs reside in haloes lower than the reliable mass threshold 6 × 10 10 ℎ −1 M ⊙ (Section 2.3).So the influence of the limitation in the -body simulation halo finder on our SHAM study can be dismissed.
The PDF of (sub)halo  peak that can host a central (satellite) galaxy is shown in Fig. 13 for all three tracers at 0.8 <  < 1.1.This is the empirical galaxy-halo relation that we calibrate using SHAM described in Section 3.2, i.e., LRG with Gaussian  smear,G , ELG and QSO.The shape of the PDF is modulated indirectly by  and  ceil .ELGs mainly reside in (sub)haloes with  peak ≲ 200 km s −1 , while LRG and QSOs are populated in haloes with  peak > 200 km s −1 .The PDFs of LRGs and ELGs present a clear peak, while the probability of QSOs in the massive end decays slower than them.The probability patterns for different tracers can be used as a reference for future multi-tracer studies.The mean  peak values for the total LRG, ELG and QSO samples are presented in Table 4.

CONCLUSIONS
We have generated catalogues of mock galaxies matching the clustering of dark tracers from the DESI One Percent Survey in the range of 5-30 ℎ −1 Mpc.The DESI samples studied here are luminous red galaxies (LRG) at 0.4 <  < 1.1; emission line galaxies (ELG) at 0.8 <  < 1.6 and QSO at 0.8 <  < 3.5 (Section 2.1).Mock galaxies are painted on the dark matter only UNIT simulation (Section 2.3), using two SubHalo Abundance Matching (SHAM) algorithms (Section 3.2).The first algorithm,  smear -SHAM, is used for LRGs and QSOs, and has the following free parameters: , to model the dispersion in the galaxy-halo mass relation but also include the incompleteness of halo mass;  ceil , to account for the incompleteness of massive haloes for the galaxy samples; and  smear , to model the uncertainty in the redshift determination process.The other SHAM model,  sat -SHAM with a free satellite fraction  sat , is introduced here to model ELGs as  sat is crucial to recover the quadrupole of DESI ELGs.The redshift uncertainty of ELGs is the lowest among the considered tracers and its  smear has a negligible impact on the clustering of SHAM ELGs down to 5 ℎ −1 Mpc.So  smear is not included in  sat -SHAM.
For LRGs, we find the best-fitting  to be consistent with 0 at a 2- level.However, ELG and QSO samples constrain  weakly.Although the loose constraint from the QSO sample is mostly due to its small number density, this does not stand for ELGs which are over 10 times denser than QSOs.We attribute this lack of constraint to the fact that  also models the incompleteness in both stellar mass and luminosity, resulting in a complex galaxy-halo relation that is harder to constrain.
ceil , the massive-(sub)halo incompleteness, describes the stellar mass incompleteness in the massive end due to both target selection criteria of galaxy surveys and the intrinsic properties of certain galaxies.The best-fitting  ceil for LRGs, is as small as 0.02 per cent.This small value shows that DESI LRGs from the One Percent Survey are close to complete at the massive end.The best fit  ceil for ELGs shows that up to 7 per cent of (sub)haloes that have the largest  scat (Eq.( 13)) in the UNIT simulation would not host ELGs.Although QSOs are the brightest object at  > 1, their best-fitting  ceil is inconsistent with zero, suggesting that not all haloes above a certain mass will be hosting a QSO.Their absence in the centre of massive (sub)haloes is consistent with the depletion of cold gas in this hot and dense environment there, leading to the quenching of ELG star formation and QSO black hole accretion.This agrees with the scenarios found in SAM studies and observations (e.g., Uchiyama et al. 2018;Griffin et al. 2019;Gonzalez-Perez et al. 2020).
smear quantifies the effect that the redshift uncertainty has on the clustering.It can also be measured statistically and independently by the redshift difference Δ of repeat observations (see Section 2.2).The Δ histogram of DESI tracers follows, in general, a Lorentzian profile with width  Δ , instead of a single Gaussian profile as was found for BOSS/eBOSS galaxies.Thus, we have developed SHAM algorithms with a truncated Lorentzian profile for modelling the redshift uncertainty, i.e.,  smear,L .The Δ of LRG sub-samples in different redshift bins can be fitted well by a Lorentzian or a Gaus-sian profile and thus, we also develop a Gaussian  smear,G for LRGs.The Lorentzian  smear,L of LRGs is only consistent with  Δ at 0.4 <  < 1.1.The clustering of LRG sub-samples in redshift bins actually suggests a preference for a Gaussian profile for the redshift uncertainty.Nevertheless, truncated Lorentzian and Gaussian functions provide the same SHAM clustering and consistent best-fitting  and  ceil .For QSOs,  smear,L monotonically increases and deviates from  Δ at  > 1.5.This is because the C iv line used to determine QSO redshifts is affected by the velocity shifts of spectral lines that can vary between objects.Although the repeat observation cannot capture this feature, its effect on the clustering will be modelled by SHAM  smear,L .This is consistent with the eBOSS QSO analysis (Zarrouk et al. 2018).
The satellite fraction  sat of LRG and QSO samples is fixed to the number of subhaloes from the UNIT simulation included for the SHAM, given  and  ceil .Their  sat decreases with redshift, following the evolution of the subhalo fraction in the simulations.For ELGs we use the  sat -SHAM, setting  sat as a free parameter.The best-fitting  sat for DESI ELGs is around 4 per cent.This low value is consistent with previous studies for strong [O ii] emitters (Gonzalez-Perez et al. 2020;Gao et al. 2022), but it is lower than the estimations for the total ELG samples (Favole et al. 2016;Guo et al. 2019;Lin et al. 2023).
We provide the halo occupation distribution (HOD) measured from our best-fitting SHAM for LRGs, ELGs and QSOs from the One Percent Survey.The HOD of SHAM central LRGs reaches its peak at ⟨⟩ = 0.75±0.07and is consistent with an incomplete LRG pattern as found in (Yuan et al. 2023).The HOD for central SHAM ELGs is consistent with a star-forming HOD profile peaking at ⟨ cen ⟩ = 0.06 ± 0.03 and  vir = 10 11.7 ℎ −1 M ⊙ , but we cannot exclude a Gaussian shape (e.g Avila et al. 2020).The HOD for SHAM central QSO also decreases after  vir = 10 12.4 ℎ −1 M ⊙ with ⟨⟩ = 0.016 ± 0.001.The HOD for all types of SHAM satellite galaxies is composed of two exponential functions with different slopes.The slope  in the massive halo end is  LRG = 0.70 +0.01 −0.01 ,  ELG = 0.76 +0.06 −0.07 , and  QSO = 0.73 +0.07 −0.07 .They are smaller than the measurements from AbacusSummit HOD tests for LRGs but consistent with those from ELGs and QSOs (Yuan et al. 2023;Rocher et al. 2023).We shall point out that the decreasing halo occupation of centrals with respect to the halo mass for LRGs, ELGs and QSOs is a result of the simple  scat truncation, i.e., the implementation of  ceil .Galaxy clustering produced by this profile is consistent with that from HOD models with other profiles.So we need more data with higher accuracy and physical models in SHAM/HOD to give a better description of this halo occupation incompleteness on the massive end.The crossvalidation of the halo occupation number among hydrodynamical simulations, SAM, forward modelling, SHAM and HOD is planned for future work.This is because we have yet a consistent clustering measurement with the observation for all methods and the series of mock galaxies generated by those methods on the same simulation.
We measure a mean parent halo mass of ⟨ vir ⟩ = 10 13.16±0.01ℎ −1 M ⊙ for LRGs, 10 11.90±0.06ℎ −1 M ⊙ for ELGs and 10 12.66±0.45ℎ −1 M ⊙ for QSOs.For sub-samples at redshift bins, we obtain ⟨ vir ⟩ that decreases with redshift for LRGs and QSOs, but not for ELGs.Meanwhile, the linear bias for each tracer increases with the redshift.Those results are consistent with the HOD measurement using the same tracers from the One Percent Survey in general.
SHAM algorithms that include the redshift uncertainty, massive-(sub)halo incompleteness and an adjustable satellite fraction work well in the single-tracer case, which can provide galaxy mocks for cosmological tests (e.g., Su et al. 2022).We plan to enhance this study in the future by implementing a multi-tracer SHAM method based on what we have learned from the present study.The impact of  (first column),  ceil (second column),  smear (third column),  sat (fourth column) on the 2PCF monopole (first row), quadrupole (second row) and the projected 2PCF (third row).The standard sample is ,  ceil ,  smear = 0 with  sat = 10.9 per cent in black lines.The first three columns show the 2PCF difference between the standard sample and samples with  = 0.5,  ceil = 0.2 per cent or  smear,L = 300 km s −1 (truncated at 2000 km s −1 ) respectively while fixing the other two parameters, in red lines.Results of  smear -SHAM with a Gaussian profile  smear,G = 80 km s −1 are also presented in the third column in blue lines.In the last column, we compare the 2PCF for SHAM galaxies with  sat = 15 per cent (red lines) and  sat = 4 per cent (blue lines) with that of the standard sample in  sat -SHAM with ,  ceil ,  smear = 0. consistent with  Δ , resolving the discrepancy shown in Figure 6 of Yu22.The  smear - sat degeneracy enables  smear to decrease to the observed uncertainty level by increasing its satellite fraction.Meanwhile, our best-fitting  sat is consistent with the LOWZ HOD  sat = 12 ± 2 per cent (Parejko et al. 2013).

APPENDIX D: IS SATELLITE FRACTION BIASED?
To validate our satellite fraction in  sat -SHAM measurement, we use other galaxy mocks as observations.They are constructed in various galaxy-halo models and have the same definition of satellites as our SHAM, i.e., galaxies residing in subhaloes.By implementing  sat -SHAM on the same -body simulations as those model galaxies and fitting the 2PCF monopole and quadrupole of those modelled galaxies on 5-30 ℎ −1 Mpc,  sat of the best-fitting SHAM is expected to be consistent with the true value of those galaxy mocks.The covariance matrices we used here are calculated with jackknife subsamples of other mock galaxies produced by pycorr.
The first set of galaxies is from a DESI-like ELG catalogue es-  tablished with SAM (Gonzalez-Perez et al. 2014, 2020, SAM ELG hereafter) at redshift  = 0.99 with  sat = 4.14 per cent.The corresponding -body simulation is Millennium 9 with the WMAP7 cosmology (Springel et al. 2005).As there is no  peak in this simulation, we use  max for  sat -SHAM.Another sample of star-forming galaxies from UniverseMachine (Behroozi et al. 2019, UniverseMachine ELG hereafter) at  = 0.9436 with star-formation rate larger than 10 1.1 M ⊙ yr −1 and 10 9.6 <  * < 10 11 M ⊙ .This galaxy catalogue is based on the snapshot of MultiDark MDPL2 simulation (Prada et al. 2012) at the same redshift 10 .Orphan galaxies (Campbell et al. 2018;Behroozi et al. 2019) are removed from this sample to ensure a fair comparison to our SHAM-reproduced results as Mul-tiDark simulations do not include this reconstruction on subhaloes by default.Finally, the star-forming galaxies from UniverseMachine are downsampled to have  gal = 10.28 × 10 −4 Mpc −3 ℎ 3 , 1.8 per cent smaller than that of DESI ELGs at 0.8 <  < 1.1.Fig. C5 shows the clustering of SAM ELGs (blue dots with error bars), UniverseMachine ELGs (magenta dots with error bars) and DESI ELGs (black dots with error bars), and the clustering of the best-fitting SHAM galaxies (lines with corresponding colours).Our  sat -SHAM can describe the 2PCF multipoles for SAM ELGs and UniverseMachine ELGs and reproduce their projected 2PCF on 5-30 ℎ −1 Mpc.The best-fitting  sat for SAM ELGs is 11.5 +1.46 −1.19 per cent, which is a 6- overstimation.Meanwhile, that for UniverseMachine ELGs is 12.60 +0.76  −0.82 per cent, consistent with the true value.The inconsistency in the estimation of the satellite fraction among different galaxy-halo models needs further discussion in future studies.

Figure 1 .
Figure 1.The statistical redshift uncertainty estimated with the histogram of the redshift difference (black filled circles with error bars) from repeat observation taken during the early stage of Survey Validation.The first, second and third rows are ELGs, LRGs and QSOs respectively.The first columns of all rows are results for total samples, while the rest are for sub-samples at redshift bins.The statistical redshift uncertainty measured by Lorentzian functions  Δ (solid red lines) and standard deviations σΔ of Δ is presented in the label of each subplot.For LRG samples, we also fit Δ histograms with Gaussian functions (blue solid lines), providing their best-fitting dispersion  Δ in the label as well.The fraction of Δ that are not included in the fittings is indicated as outlier fractions in titles.

Figure 2 .
Figure 2. The clustering of observed LRGs (filled circles with error bars) compared with that of the best-fitting SHAM model galaxies with its statistical uncertainty (solid lines with shades).Monopoles and their residuals normalised by the observed errors  obs are presented in the first and the second rows.The third and fourth rows present those for quadrupoles.Each colour shows a different redshift range, as indicated in the legend.The first column shows results for LRGs in different redshift bins and the second for the total sample.The error bars of data are obtained from 128 jackknife samples, and the statistical uncertainty of SHAM galaxies indicated in the width of the shades is the standard deviation of its 32 realizations divided by

Figure 6 .Figure 7 .
Figure6.The redshift uncertainty quantified by the best-fitting SHAM  smear (filled stars with error bars, the subscripts 'L' and 'G' stand for Lorentzian and Gaussian profiles) and that estimated statistically by the repeat observation Δ (empty squares with error bars,  Δ for Lorentzian and  Δ for Gaussian) for galaxies in different redshift slices.The results of Lorentzian profiles are in red colours and those of Gaussian profiles are in blue.For DESI LRGs, SHAM  smear,L are systematically lower than  Δ .In the case of the Gaussian profile, the SHAM  smear,G and  Δ (both with vertical offsets) agree with each other.For QSOs, the statistical uncertainty  Δ is not consistent with the SHAM  smear,L (black filled stars with error bars) at  > 1.5.

Figure 8 .
Figure 8.  sat evolution with  (left panel),  ceil (middle panel) and  eff (right panel) for model galaxies of SHAM at  = 0.94 (red empty error bars) and those at  = 1.83 (blue empty error bars) produced by  smear -SHAM.We present the dependence of  sat with one parameter and fix the other two parameters (typical values for LRGs) as indicated in the label.The error bar of  sat is the standard deviation of  sat among 32 realizations.The subhalo fraction of the UNIT simulation at  = 0.94 (red) and  = 1.83 (blue) defined in Eq. (17) is plotted in dashed lines.

Figure 9 .
Figure 9.The flux (in erg cm −2 s −1 ) and the luminosity (in erg s −1 ) of [O ii] emission for ELGs as a function of the redshift.The red lines are the strong [O ii] emitter threshold.The line on the left represents the standard of Gonzalez-Perez et al. (2018) with  [O ii] > 10 −16 erg s −1 cm −2 .The one on the right shows an evolving  thres [O ii] (Eq.(24)) derived from Gao et al. (2022).71 per cent of DESI ELGs pass the  [O ii] selection, 24 per cent pass the  [O ii] selection.

Figure 11 .Figure 12 .
Figure 11.The average HOD of SHAM LRGs at 0.4 <  < 1.1 (left panel), ELGs at 0.8 <  < 1.6 (middle panel) and QSOs at 0.8 <  < 3.5 (right panel).The contribution of central galaxies to the average HOD is shown by red lines and that of satellites by blue lines.The 1- errors derived from the Monte-Carlo chain are shown as shaded regions of the same colour.
Figure A1.The impact of  (first column),  ceil (second column),  smear (third column),  sat (fourth column) on the 2PCF monopole (first row), quadrupole (second row) and the projected 2PCF (third row).The standard sample is ,  ceil ,  smear = 0 with  sat = 10.9 per cent in black lines.The first three columns show the 2PCF difference between the standard sample and samples with  = 0.5,  ceil = 0.2 per cent or  smear,L = 300 km s −1 (truncated at 2000 km s −1 ) respectively while fixing the other two parameters, in red lines.Results of  smear -SHAM with a Gaussian profile  smear,G = 80 km s −1 are also presented in the third column in blue lines.In the last column, we compare the 2PCF for SHAM galaxies with  sat = 15 per cent (red lines) and  sat = 4 per cent (blue lines) with that of the standard sample in  sat -SHAM with ,  ceil ,  smear = 0.

Figure C1 .
Figure C1.The projected 2PCFs     of observed LRGs (filled circles with error bars) and those of the best-fitting SHAM galaxies (solid lines with shades) with  max = 30 ℎ −1 Mpc.The direct comparison and residuals rescaled by the observed error bars are presented in the first and the second row respectively.The samples on the left are LRGs at different redshift bins, while the sample on the right is the total sample.

Figure C5 .
Figure C5.The 2PCF of UniverseMachine ELGs (magenta circles with error bars) at  = 0.9436, SAM ELGs (blue circles with error bars) at  = 0.99, and DESI ELGs (black circles with error bars) with  eff = 0.9565, and their corresponding best-fitting SHAM galaxies.The 2PCF monopoles, quadrupole and projected 2PCF are presented in the left, middle and right panels respectively.

Table 1 .
The priors of  smear -SHAM for LRGs and QSOs, and of  sat -SHAM for ELGs.Priors of  smear -SHAM with a Gaussian profile  smear,G are the same with those with  smear,L .

Table 2 .
The information for observation and its best-fitting SHAM results of  smear -SHAM with Lorentzian redshift uncertainty profile  smear,L and  sat -SHAM.

Table 3 .
The best-fitting results of the  smear -SHAM fitting for LRGs with Gaussian  smear,G .