The DESI One-Percent Survey: A Concise Model for the Galactic Conformity of Emission-line Galaxies

DESI is a Stage IV dark energy survey that aims to collect the spectra of about 40 million extragalactic objects over 5 yr (Levi et al. 2013; DESI Collaboration et al. 2016a, 2016b, 2022). This survey will cover more than 14,000 deg² in sky and has been conducted using a multi-object fiber-fed spectrograph mounted on the prime focus panel of the 4 m Mayall Telescope situated in the Kitt Peak National Observatory (DESI Collaboration et al. 2022). The spectrometer has a wavelength range of 3600–9800 Å and can allocate fibers to 5000 objects at one visit (DESI Collaboration et al. 2016b; Miller et al. 2023; Silber et al. 2023). Several auxiliary pipelines supporting the DESI experiment are described in S. Bailey et al. (2023, in preparation), Guy et al. (2023), Myers et al. (2023), A. Raichoor et al. (2023, in preparation), and Schlafly et al. (2023). The strategy of the DESI target selection and survey validation has been presented in a series of works (Allende Prieto et al. 2020; Raichoor et al. 2020, 2023; Ruiz-Macias et al. 2020; Yèche et al. 2020; Zhou et al. 2020, 2023; Alexander et al. 2023; Chaussidon et al. 2023; Cooper et al. 2023; Hahn et al. 2023; Lan et al. 2023).

As the third phase of the survey validation (DESI Collaboration et al. 2023a), the One-Percent Survey is used to further optimize and verify the efficiency of the observation and manipulation procedures. Its footprint covers 20 discrete small regions of the sky, each covering an area of about 7 deg². The One-Percent Survey contains a total of 488 tiles, consisting of ∼10–11 (∼12–13) repeated visits in bright (dark) time for each field. As the result, the efficiency of the fiber placement and the success rate of the spectroscopic measurements are quite high. More than 99% of LRGs and 95% of ELG samples can be successfully assigned to fibers. Consequently, galaxy samples in the One-Percent Survey suffer little from fiber collisions, allowing the measurement of clustering to be extended to smaller scales (<0.1 Mpc h⁻¹).

The DESI Early Data Release (DESI Collaboration et al. 2023b) provides both the full and the clustering catalogs for the One-Percent Survey. We take the ELG and LRG samples with successful redshift measurements from the clustering catalog. For each galaxy, with five-band photometry grzW1W2 (The Dark Energy Survey Collaboration 2005; Wright et al. 2010; Zou et al. 2017; Dey et al. 2019, D. J. Schlegel et al. 2023, in preparation), we conduct a spectral energy distribution fit using CIGALE (Boquien et al. 2019). See Paper I for detailed template and model settings. Using exactly the same approach as in Paper I, we select four ELG subsamples (ELG0, ELG1, ELG2, and ELG3) and three LRG subsamples (LRG0, LRG1, and LRG2), binned by stellar mass. We measure the auto correlation functions of the different subsamples and the cross correlations between the ELG and LRG subsamples. In addition, we also select the entire ELG sample (without stellar mass bins) at different redshifts in our analysis. The details of our galaxy samples are shown in Table 1. In the measurements of the entire ELG samples, we correct the fiber collision by combining the completeness weight derived from the 128 merged target list realizations (J. Lasker et al. 2023, in preparation), the pairwise-inverse-probability (PIP) weight (Bianchi & Percival 2017), and the angular up-weight (ANG; Percival & Bianchi 2017; Mohammad et al. 2020). The combination of these weights can increase the correlation function by ∼10% at ∼0.1 Mpc h⁻¹.

Table 1. Details of the Galaxy Samples Used in Our Analysis

Name	Redshift Range	$\mathrm{log}{M}_{* }({M}_{\odot })$	Ng
LRG0	0.8 < z ≤ 1.0	$\left[11.1,11.3\right]$	13,906
LRG1	0.8 < z ≤ 1.0	$\left[11.3,11.5\right]$	4834
LRG2	0.8 < z ≤ 1.0	$\left[11.5,11.7\right]$	957
ELG0	0.8 < z ≤ 1.0	$\left[8.5,9.0\right]$	9481
ELG1	0.8 < z ≤ 1.0	$\left[9.0,9.5\right]$	29,764
ELG2	0.8 < z ≤ 1.0	$\left[9.5,10.0\right]$	34,155
ELG3	0.8 < z ≤ 1.0	$\left[10.0,10.5\right]$	6583
All ELG	0.8 < z ≤ 1.0	⋯	82,887
All ELG	1.0 < z ≤ 1.2	⋯	68,366
All ELG	1.2 < z ≤ 1.4	⋯	54,783
All ELG	1.4 < z ≤ 1.6	⋯	37,756

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We adopt the classic Landy–Szalay estimator (Landy & Szalay 1993; Szapudi & Szalay 1998) to estimate the two-point correlation function of subsamples x and y:

$$\begin{eqnarray}&&{\xi }_{{xy}}\left({r}_{{\rm{p}}},{r}_{\pi }\right)=\left[\displaystyle \frac{{D}_{x}{D}_{y}-{D}_{x}{R}_{y}-{D}_{y}{R}_{x}+{R}_{x}{R}_{y}}{{R}_{x}{R}_{y}}\right]\left({r}_{{\rm{p}}},{r}_{\pi }\right),\end{eqnarray} \tag{ 1 }$$

where D_x D_y , D_x R_y , D_y R_x , and R_x R_y denote the normalized weighted pair counts in each $\left({r}_{{\rm{p}}},{r}_{\pi }\right)$ bin. In the measurement, we set 25 r_p bins from 10^−1.5 to 10^1.477 Mpc h⁻¹ in logarithmic space and 40 r_π bins from 0 to 40 Mpc h⁻¹ in linear space.

We then integrate the ${\xi }_{{xy}}\left({r}_{{\rm{p}}},{r}_{\pi }\right)$ along the direction of r_π (Davis & Peebles 1983):

$$\begin{eqnarray}&&{w}_{{\rm{p}},{xy}}\left({r}_{{\rm{p}}}\right)=2{\int }_{0}^{{r}_{\pi ,\max }}{\xi }_{{xy}}\left({r}_{{\rm{p}}},{r}_{\pi }\right){\rm{d}}{r}_{\pi },\end{eqnarray} \tag{ 2 }$$

where ${w}_{{\rm{p}},{xy}}\left({r}_{{\rm{p}}}\right)$ is the projected correlation function in real space and ${r}_{\pi ,\max }=40\,\mathrm{Mpc}\,{h}^{-1}$ is the upper limit of the integration.

Moreover, the multipole moments in redshift space can be calculated through (Hamilton 1992)

$$\begin{eqnarray}&&{\xi }_{l,{xy}}\left(s\right)=\displaystyle \frac{2l+1}{2}{\int }_{-1}^{1}{\xi }_{{xy}}\left(s,\mu \right){L}_{l}\left(\mu \right){\rm{d}}\mu ,\end{eqnarray} \tag{ 3 }$$

where $s=\sqrt{{r}_{{\rm{p}}}^{2}+{r}_{\pi }^{2}}$ and μ = r_π /s. The monopole, quadrupole, and hexadecapole correspond to l = 0, 2, and 4, respectively. We adopt 15 s bins from 0.3 to 30 Mpc h⁻¹ in logarithmic space and 10 μ bins from 0 to 1 in linear space.

We divide the area of the One-Percent Survey into 100 approximately equal regions and estimate the covariance matrix of w _p and ξ _l using a jackknife technique.

The N-body simulation used in this study is from CosmicGrowth (Jing 2019). The box size of this simulation is 600 Mpc h⁻¹ and the number of particles is 3072³. The simulation sets the standard ΛCDM cosmological parameters —Ω_m,0 = 0.268, Ω_Λ,0 = 0.732, h=0.71, n_s = 0.968, and σ₈ = 0.83—and it was run with an adaptive P³M algorithm (Jing & Suto 2002). The achieved mass resolution is m_p = 5.54 × 10⁸ M_⊙ h⁻¹.

The halo catalog is identified using the friends-of-friends algorithm (Davis et al. 1985). We define the virial mass ${M}_{\mathrm{vir}}=\tfrac{4}{3}\pi {R}_{\mathrm{vir}}^{3}{{\rm{\Delta }}}_{\mathrm{vir}}$ as the default halo mass M_h, where Δ_vir is the overdensity for a virialized spherical structure (Bryan & Norman 1998). The subhalo catalog and merger trees are constructed through the Hierarchical-Bound-Tracing algorithm (HBT+; Han et al. 2012, 2018). We define the default mass M_s of the subhalo as the virial mass at the last moment before infall. We further calculate the merging timescale of the subhalos with particle number less than 20 to determine whether they are still alive (Jiang et al. 2008). The final halo and subhalo mass functions are consistent with the theoretical expectations (Jing 2019; Xu et al. 2022).

As in Paper I, the redshift space distortion is added along the z-direction for each halo and subhalo. When modeling the galaxy clustering in redshift space, we assume a Gaussian scatter σ_z = 10 km s⁻¹ for ELGs and σ_z = 40 km s⁻¹ for LRGs to account for the redshift uncertainties. Besides, we also incorporate the velocity bias (e.g., Yoshikawa et al. 2003) for the central galaxy using a random Gaussian distribution with σ_c = α_c × σ_v , where ${\sigma }_{v}=\sqrt{{{GM}}_{\mathrm{vir}}/\left(2{R}_{\mathrm{vir}}\right)}$ is the one-dimensional velocity dispersion of a halo and α_c is fixed to 0.22 (Guo et al. 2015).

We take four snapshots at z = 0.92, 1.09, 1.27, and 1.47 to cover the redshift range of 0.8 < z < 1.6 in DESI.

The basic idea of the original SHAM method developed by Gao et al. (2022) and Paper I is to first obtain normal galaxies in dark matter halos, and then to reduce the satellite fraction in massive halos to get ELG candidates. The final ELG sample can be randomly selected from the candidates by matching the observed stellar mass function (SMF) of the ELG.

3.1.1. Populate Halo (Subhalo) with Normal Galaxies

Given a halo or subhalo in the N-body simulation, we can assign a stellar mass M_* to it through the stellar–halo mass relation (SHMR) model:

$$\begin{eqnarray}&&p({M}_{* }| {M}_{{\rm{h}}})=\displaystyle \frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\displaystyle \frac{{\left(\mathrm{log}{M}_{* }-\mathrm{log}\left\langle {M}_{* }| {M}_{{\rm{h}}}\right\rangle \right)}^{2}}{2{\sigma }^{2}}\right],\end{eqnarray} \tag{ 4 }$$

where M_h is the mass of the halo (subhalo) and the intrinsic dispersion of the SHMR is quantified by a Gaussian scatter σ. The mean relation of M_* and M_h can be formulated by a double-power-law function (Wang et al. 2006; Wang & Jing 2010; Yang et al. 2012; Moster et al. 2013):

$$\begin{eqnarray}&&\left\langle {M}_{* }| {M}_{{\rm{h}}}\right\rangle =\displaystyle \frac{2k}{{\left({M}_{{\rm{h}}}/{M}_{0}\right)}^{-\alpha }+{\left({M}_{{\rm{h}}}/{M}_{0}\right)}^{-\beta }},\end{eqnarray} \tag{ 5 }$$

where the parameters α and β denote the slopes at the high- and low-mass ends, respectively, M₀ is the dividing point of the two power laws, and the constant k is used for normalization.

Once the SHMR is applied to all the halos and subhalos, we can generate galaxies with stellar masses in the simulation. Using the best-fit SHMR (see the first five columns of Table 2) provided by Paper I, we place galaxies in each halo and subhalo in the simulation. The number densities of these central ${n}_{\mathrm{cen}}^{\mathrm{nor}}$ and satellite galaxies ${n}_{\mathrm{sat}}^{\mathrm{nor}}$ are shown as the shaded lines in the top left panel of Figure 1. These galaxies are complete in terms of stellar mass, free from any selection effects, and therefore are called normal galaxies. As shown by the SMFs in Paper I, the LRG and ELG samples in DESI are subject to complex selection effects. LRGs are predominantly massive red galaxies and can be considered nearly complete only at the massive end (>10^11.3 M_⊙). ELGs make up only a small fraction (<10%) of the total normal galaxy population at each stellar mass and are biased against satellite galaxies in massive halos. In the next subsection, we will mimic the selection functions of LRGs and ELGs and select them from the normal galaxies.

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Table 2. All Parameters in the Model

$\mathrm{log}{M}_{0}\left({M}_{\odot }\,{h}^{-1}\right)$	α	β	$\mathrm{log}k$	σ	a	b	c	${K}_{\mathrm{conf}}\ \left(\mathrm{This}\,\mathrm{Work}\right)$
12.07	0.37	2.61	10.36	0.21	1.00 (fixed)	12.55	0.04	${5.21}_{-0.50}^{+0.54}$

Note. The first eight parameters correspond to the SHMR and P_sat models in Paper I. Except for the parameter a that is fixed to 1, the remaining seven parameters are the best-fit values. The last one is the best-fit conformity parameter K_conf in this work.

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3.1.2. Select ELGs from Normal Galaxies

Next, we can select ELGs from the normal galaxies. The selection can be further divided into two steps.

The first step is to select ELG candidates. As demonstrated in Paper I, relative to a central galaxy, we need to reduce the probability P_sat that a satellite galaxy can be selected as an ELG candidate. The physical motivation for this hypothesis is that the star formation of satellite galaxies may be quenched by the dense environment, reducing the fraction of satellite ELGs. In Paper I, we proposed a halo-mass-dependent model for P_sat:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\mathrm{sat}}({M}_{{\rm{h}}})=\displaystyle \frac{a}{2}\times \left[1-\mathrm{erf}(\mathrm{log}{M}_{{\rm{h}}}-b)\right]\\ +\displaystyle \frac{c}{2}\times \left[1-\mathrm{erf}(b-\mathrm{log}{M}_{{\rm{h}}})\right].\end{array}\end{eqnarray} \tag{ 6 }$$

This model reduces the number of satellite ELGs in the massive halos, while retaining satellite ELGs in the low-mass halos. In this way, we keep all the normal central galaxies and randomly select some normal satellite galaxies as ELG candidates according to their probability P_sat. At a fixed stellar mass, the satellite fraction of the ELG candidates is identical to the final ELG mock samples we want to get, and the only difference between the two samples is the number density. In the top left panel of Figure 1, we present the number densities ${n}_{\mathrm{cen}}^{\mathrm{can}}$ and ${n}_{\mathrm{sat}}^{\mathrm{can}}$ of ELG candidates as a function of stellar mass. We can notice that the fraction of normal satellite galaxies selected as ELG candidates gradually decreases as the stellar mass increases.

The second step is to select the ELGs. In this selection, we only need to adjust the number density of galaxies in the simulation to match the ELG SMF in the observations. It is effectively a downsampling process. We set 45 bins in logarithmic spaces from 10^7.5 to 10¹² M_⊙ h⁻¹ with equal widths of ${\rm{\Delta }}\mathrm{log}{M}_{* }=0.1$. In each bin, we calculate the mean number density of ELGs ${\bar{n}}_{\mathrm{obs}}^{\mathrm{ELG}}\left({M}_{* }\right)$ in observations (See Paper I for the SMF of ELGs) and ELG candidates ${\bar{n}}_{\mathrm{can}}^{\mathrm{ELG}}\left({M}_{* }\right)$ in simulation. Based on the ratio between the two number densities, we can define a probability ${F}^{\mathrm{ELG}}\left({M}_{* }\right):$

$$\begin{eqnarray}&&{F}^{\mathrm{ELG}}\left({M}_{* }\right)=\displaystyle \frac{{\bar{n}}_{\mathrm{obs}}^{\mathrm{ELG}}\left({M}_{* }\right)}{{\bar{n}}_{\mathrm{can}}^{\mathrm{ELG}}\left({M}_{* }\right)}.\end{eqnarray} \tag{ 7 }$$

For each ELG candidate in the simulation, we can assign a probability ${F}^{\mathrm{ELG}}\left({M}_{* }\right)$ for it. A random selection is then performed to pick up the final ELG sample in the simulation. The stellar mass distributions of the selected ELGs ${n}_{\mathrm{cen}}^{\mathrm{ELG}}$ and ${n}_{\mathrm{sat}}^{\mathrm{ELG}}$ are presented in the top right panel of Figure 1 as solid lines.

Our model has a total of eight parameters {α, β, M₀, k, σ, a, b, c}, but there are only seven free parameters, as a is set to 1 in Paper I. Using the auto and cross correlation functions of ELGs and LRGs, Paper I simultaneously constrained the SHMR for normal galaxies and the P_sat model. The best-fit parameters of Paper I are listed in the first eight columns of Table 2.

Taking these parameters, we make a model prediction for the projected correlation functions w _p shown in Figure 2. We can see that the model is sufficient to reproduce the LRGxELG cross correlation functions. But at r_p < 0.3 Mpc h⁻¹, the observed ELG auto correlation is clearly much higher than the prediction of the model. We note that Paper I has fixed the parameter a to 1, which means no reduction of the ELG satellite fractions in typical halos of M_h ≈ 10¹² M_⊙ h⁻¹ that host central ELGs. Namely, the ELG satellite fraction in these halos is the same as that of normal galaxies. Nevertheless, it is still difficult for the model to match the observed ELG auto correlation on small scales. Therefore, we consider incorporating the conformity effect in the original SHAM approach.

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We aim to propose a concise empirical model of ELG conformity. As argued in the previous subsection, without conformity, we are unable to reproduce the strong auto correlation of ELGs observed on small scales (r_p < 0.3 Mpc h⁻¹), even if the satellite probability P_sat is already close to 1 for relevant halos of mass M_h < 10¹² M_⊙ h⁻¹. This fact indicates that we need to include some sort of effect, such as galactic conformity, to increase the number of close pairs. As a working hypothesis, we boost the number of satellite ELGs in those halos that host central ELGs (hereafter, central ELG halos).

First, we implement the original SHAM method, as described in section 3.1.2 (see also the top two panels in Figure 1). In this way, we obtain the ELG central and satellite galaxies in the simulation. For the central ELG halos, we label the number densities of satellite ELGs and of satellite ELG candidates as ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}$ and ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{can}}$, respectively, and plot them as the green solid and dashed curves in the bottom left panel of Figure 1. Compared to the total number density ${n}_{\mathrm{sat}}^{\mathrm{ELG}}$ of satellite ELGs, the fraction of the population in central ELG halos is quite small (less than 3%), as only a small fraction of central galaxies are qualified for observed ELGs. Comparing the solid and dashed green curves, we find that in the central ELG halos, there are still many satellite ELG candidates that were not selected, because only a small fraction is supposed to be included in the observation (see Equation 7).

To fit the one-halo term of the ELG auto correlation, we should boost the number density ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}$ to increase the central–satellite and satellite–satellite pair counts of ELGs. To achieve this, we can convert more satellite ELG candidates into final ELGs. Here we introduce K_conf as a free parameter that controls the conversion. At a given stellar mass M_*, we boost the number of the satellite ELG population in the central ELG halos by a factor of K_conf over the standard satellite occupation number, up to the limit allowed by the remaining satellite ELG candidates. The boosted number density of ELGs ${n}_{\mathrm{sat}}^{\mathrm{conf}}\left({M}_{* }\right)$ can be written as

$$\begin{eqnarray}&&\begin{array}{l}{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{conf}}\left({M}_{* }\right)=\min \{{K}_{\mathrm{conf}}\times {n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}\left({M}_{* }\right),\\ \left[{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{can}}\left({M}_{* }\right)-{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}\left({M}_{* }\right)\right]\},\end{array}\end{eqnarray} \tag{ 8 }$$

Since the number of remaining candidates is finite, we take the minimum value between ${K}_{\mathrm{conf}}\times {n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}$ and $\left[{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{can}}-{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}\right]$ as the final ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{conf}}$. Eventually, we can randomly select galaxies with the number density of ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{conf}}\left({M}_{* }\right)$ from the remaining candidates to augment the satellite ELG population in the central ELG halos.

In the bottom right panel of Figure 1, we set K_conf = 10 and illustrate the above process. We observe that the number density of the satellite ELG population in the central ELG halos, (${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{ELG}}+{n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{conf}}$), is significantly boosted. In particular, the number density of galaxies has reached the upper limit at stellar mass ∼10^9.5 M_⊙. At the low- and high-mass ends, there is still ample room for increasing ${n}_{\mathrm{sat},\mathrm{CE}}^{\mathrm{conf}}$. Moreover, it should be pointed out that this process can cause the ELG SMF in the simulation to be slightly higher, by less than 2%. Nevertheless, as we will show in the next section, the impact of this systematic on the large-scale clustering is also negligible compared to the current measurement accuracy.

In this section, we aim to determine the free parameter K_conf. In real space, four ELG projected auto correlations ${{\boldsymbol{w}}}_{{\rm{p}},\mathrm{ELG}}^{\mathrm{obs}}$ and 12 LRGxELG projected cross correlations ${{\boldsymbol{w}}}_{{\rm{p}},\mathrm{LRGxELG}}^{\mathrm{obs}}$ are used in the fitting process. The resulting ${\chi }_{{w}_{{\rm{p}}},{xy}}^{2}$ can be expressed as

$$\begin{eqnarray}&&\begin{array}{l}{\chi }_{{w}_{{\rm{p}}},{xy}}^{2}\\ =\,\displaystyle \sum _{k=1}^{{N}_{{r}_{{\rm{p}}}}}\displaystyle \sum _{m=1}^{{N}_{{r}_{{\rm{p}}}}}\left({w}_{{\rm{p}},{xy},k}^{\mathrm{obs}}-{w}_{{\rm{p}},{xy},k}^{\mathrm{mod}}\right){C}_{{xy},{km}}^{-1}\left({w}_{{\rm{p}},{xy},m}^{\mathrm{obs}}-{w}_{{\rm{p}},{xy},m}^{\mathrm{mod}}\right),\end{array}\end{eqnarray} \tag{ 9 }$$

where x and y indicate different subsamples (x = y for auto correlation) and ${w}_{{\rm{p}}}^{\mathrm{obs}}$ and ${w}_{{\rm{p}}}^{\mathrm{mod}}$ represent the observed and modeled correlation functions.

Similarly, we also include the clustering in redshift space in the fit. For the subsamples x and y, the corresponding ${\chi }_{{\xi }_{l},{xy}}^{2}$ is

$$\begin{eqnarray}&&{\chi }_{{\xi }_{l},{xy}}^{2}=\displaystyle \sum _{k=1}^{{N}_{s}}\displaystyle \sum _{m=1}^{{N}_{s}}\left({\xi }_{l,{xy},k}^{\mathrm{obs}}-{\xi }_{l,{xy},k}^{\mathrm{mod}}\right){C}_{{xy},{km}}^{-1}\left({\xi }_{l,{xy},m}^{\mathrm{obs}}-{\xi }_{l,{xy},m}^{\mathrm{mod}}\right),\end{eqnarray} \tag{ 10 }$$

where ξ_l denotes the lth multiple moment.

In this way, the total χ² for all combinations of subsamples can be written as

$$\begin{eqnarray}&&\begin{array}{l}{\chi }^{2}=\displaystyle \sum _{x}{\chi }_{{w}_{{\rm{p}}},{xx}}^{2}+\displaystyle \sum _{x}\displaystyle \sum _{y}{\chi }_{{w}_{{\rm{p}}},{xy}}^{2}\\ +\,\displaystyle \sum _{x}\displaystyle \sum _{l}{\chi }_{{\xi }_{l},{xx}}^{2}+\displaystyle \sum _{x}\displaystyle \sum _{y}\displaystyle \sum _{l}{\chi }_{{\xi }_{l},{xy}}^{2},\end{array}\end{eqnarray} \tag{ 11 }$$

where x ∈ [ELG0, ELG1, ELG2, ELG3], y ∈ [LRG0, LRG1, LRG2], and l ∈ [0, 2, 4]. We perform a Markov Chain Monte Carlo analysis utilizing emcee (Foreman-Mackey et al. 2013). The posterior distribution of K_conf in the parameter space is displayed in Figure 3 and the last column of Table 2.

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The fitting of all these correlation functions yields a reduced χ² = 1184.07/1076 = 1.10. In Figure 4, the best-fit ${{\boldsymbol{w}}}_{{\rm{p}}}^{\mathrm{mod}}$ models are plotted as solid lines. Compared to the top left panel of Figure 2, the model for ELG auto correlations at r_p < 0.5 Mpc h⁻¹ is greatly improved. To achieve such strong clustering, the number of satellite ELGs with respect to a central ELG should be increased by a factor of 5.21. In the case of LRGxELG cross correlations, the conformity effect has a minimal impact on the results, as expected. This is because the majority of LRGs are massive central galaxies, while the conformity effect primarily enhances the number of satellite ELGs in low-mass halos that host central ELGs. As a result, this effect has little influence on the LRGxELG cross correlations.

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In addition to ${{\boldsymbol{w}}}_{{\rm{p}}}^{\mathrm{mod}}$, we also check the best-fit correlation functions ξ ₀, ξ ₂, and ξ ₄ in redshift space, as displayed in Figures 5, 6, and 7, respectively. We also compare our model predictions with and without galactic conformity with the auto correlations of all ELGs, both in real space and in redshift space (Figure 8). With the galactic conformity, we are now able to get a good fitting result down to s ∼ 0.3 Mpc h⁻¹. Most of the deviations between the data points and the models fall within the 2σ range. The results indicate that our model agrees well with all data points from large to the smallest scales (r_p ∼ 0.03 Mpc h⁻¹ in real space and s ∼ 0.3 Mpc h⁻¹ in redshift space) where the correlations can be reasonably measured.

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Recently, Rocher et al. (2023) also carefully analyzed the conformity bias of the ELGs in the One-Percent Survey. They extended the HOD framework to include a conformity model in which satellite ELGs are only allowed to exist in the central ELG halos. Although the ELG clustering can be well described by their model, the introduction of conformity changes the original HOD parameters. Especially, the satellite fraction is reduced from ∼12% to ∼2%. By its construction, the cross correlation between ELGs and LRGs would be expected to be very low (even negative) on scales smaller than the typical LRG halo size (∼0.4 Mpc h⁻¹), which is contradictory with the cross correlation measurement (Figure 4).

In practice, for cosmological analysis, one usually measures the clustering of the full ELG sample. Therefore, we need to verify that our model is valid for the entire ELG sample, not just for the subsamples binned by stellar mass. In Figure 9, we present the evolution of the ELG auto correlations from redshift z = 0.8 to z = 1.6. The ELG in each redshift interval is the full sample without stellar mass binning. The correlation functions in both real space and redshift space are shown together in Figure 9.

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The model predictions are plotted as the dashed lines. In general, the model predictions are in very good agreement with the data. The only difference appears to be in the w _p at r_p ∼ 0.4 Mpc h⁻¹, where the one observed data point at each redshift is always below the model prediction. We notice that the drop scale moves to a larger scale as the redshift increases. If there is some angular observational effect that is not fully corrected in the correlation analysis on the scale 045, we would expect the scale of the drop to shift with the redshift. Interestingly, this scale is nearly the same as the mean separation of the fibers, and therefore we suspect that the drop may be due to uncorrected fiber collisions, even though the PIP and ANG up-weighting have been taken into account in our measurement. Apart from this specific difference, our model is in good agreement with the observational data within 2σ. This means that the model for both the ELG–halo connection and the conformity has a weak dependence on redshift. With the parameters obtained at z ∼ 0.9, we can extend the model to z ∼ 1.5 without requiring further modifications to the model parameters.

We further investigate the HOD of ELGs after introducing the conformity model. We apply the new SHAM method to the simulation and perform 1000 random realizations. The values of HOD presented in Figure 10 are the average of these realizations. We show the central occupations N_cen(M_h) as solid lines in the left panel. We can see that the N_cen peaks around 10¹² M_⊙ h⁻¹ and decreases toward the low- and high-mass ends. The position of the peak also gradually increases with redshift. The shape and evolution of the N_cen has been analyzed in detail in Paper I. By our design, the conformity model does not have any effect on the N_cen.

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As for the satellite occupations N_sat(M_h), we discuss them in two cases. First, for the halos whose central galaxies are not ELGs, we display their N_sat as solid lines in the right panel of Figure 10. In this case, the N_sat is actually equivalent to the original model without conformity effect. Next, we also present the N_sat for the central ELG halos as the dotted lines. We can notice that although the conformity model increases the number of satellite ELGs in the central ELG halos by a factor of ∼5, the N_sat is still smaller than 1, even in the massive halos with mass ∼10¹⁴ M_⊙ h⁻¹. In other words, the number of satellite ELGs is always small (<1), even in the central ELG halos. On the one hand, the P_sat model is only ∼0.04 at the massive end, and thus the number of ELG candidates in the large halo is small. On the other hand, the observed ELG SMF limits the number of ELGs that can ultimately be selected. As a result, this conformity effect does not significantly increase the overall number of satellite ELGs in the model. For example, at z ∼ 0.9, the ELG satellite fraction in the model has only changed from 15.7% to 17.6% when conformity is included.

In our model, the best-fitting conformity parameter K_conf is ${5.21}_{-0.50}^{+0.54}$, which appears to be a large effect compared to the previous results about galactic conformity (e.g., Weinmann et al. 2006). As the HODs in Figure 10 show, DESI ELGs represent less than 10% of the galaxies at a given stellar mass, and they are the bluest galaxies with the strongest [O ii] emission lines. However, previous studies usually divided galaxies into two populations, red and blue, according to their colors. This may result in a larger population of blue (or star-forming) galaxies than the DESI ELG samples. The measurements from Gao et al. (2022) also demonstrate that an ELG sample with higher [O ii] luminosity presents a stronger conformity signal than one with lower [O ii] luminosity. This explains why the DESI ELGs that are prominent [O ii] emitters show the pronounced conformity effect.