Photometric Objects Around Cosmic Webs (PAC). VI. High Satellite Fraction of Quasars

Suppose we have two populations of objects, one from a spectroscopic catalog and the other from a photometric catalog. PAC can measure the excess surface density ${\bar{n}}_{2}{w}_{{\rm{p}}}$ of photometric objects with certain physical properties around spectroscopic objects across a wide range of scales:

$$\begin{eqnarray}&&{\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})=\displaystyle \frac{{\bar{S}}_{2}}{{r}_{1}^{2}}{\omega }_{12,\mathrm{weight}}(\theta ),\end{eqnarray} \tag{ 1 }$$

where ${\bar{n}}_{2}$ is the mean number density of the photometric objects and ${\bar{S}}_{2}$ is the mean angular surface density of the photometric objects. r₁ is the comoving distance to the spectroscopic objects. w_p(r_p) and ω_12,weight(θ) are the projected cross-correlation function and the weighted angular cross-correlation function (ACCF) between the spectroscopic objects and photometric objects, with r_p = r₁ θ. To account for the case that r_p varies with r₁ at fixed θ caused by the redshift distribution of the spectroscopic objects, ω₁₂(θ) is weighted by $1/{r}_{1}^{2}$. With PAC, we can take advantage of the deep photometric surveys to statistically obtain the rest-frame physical properties without making use of photo-z. We list the key steps of PAC as follows and refer readers to Xu et al. (2022b) for a detailed account:

• 1.  
  Divide spectroscopic objects into several subsamples with narrower redshift bins to limit the range of r₁ in each bin.
• 2.  
  Assign the median redshift in each redshift bin to the entire photometric sample. The physical properties of photometric objects can be estimated through spectral energy distribution (SED) fitting with the assigned redshift. Consequently, in each redshift bin, there is a physical property catalog for the photometric sample.
• 3.  
  In each redshift bin, select photometric objects with specific physical properties and calculate ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ according to Equation (1). Foreground and background objects with incorrect redshifts are effectively eliminated through ACCF, leaving only the photometric objects around spectroscopic objects with the correct physical properties.
• 4.  
  Combine the results from different redshift bins by averaging with appropriate weights.

For the photometric sample, we utilize the DR9 catalog from the Legacy Surveys (Dey et al. 2019), which includes the Dark Energy Camera Legacy Survey (DECaLS), the Mayall z-band Legacy Survey (MzLS), and the Beijing–Arizona Sky Survey (BASS). ⁶ They cover approximately 14,000 deg² in the Northern and Southern Galactic caps (NGC and SGC), with 9000 deg² at decl. ≤ 32° and 5100 deg² at decl. > 32°. The sources are extracted using Tractor (Lang et al. 2016) and subsequently modeled with profiles convolved with a specific point-spread function. These profiles include a delta function for point sources and an exponential law, de Vaucouleurs law, and Sérsic profile for extended sources. We use their best-fit model magnitudes throughout the paper, with the Galactic extinction corrected with the maps of Schlegel et al. (1998). We merely include the footprints observed at least once in the g, r, and z bands. To remove bright stars and bad pixels, we use MASKBITS, ⁷ offered by the Legacy Surveys.

For the spectroscopic sample, we utilize the SDSS-IV eBOSS DR16 quasar sample, ⁸ which does not include SDSS-I/II/III quasars, within the redshift range 0.8 < z_s < 1.0. The sample comprises 32,291 quasars. This redshift range is selected based on the depth of the Legacy Surveys. The majority of the eBOSS quasars were targeted by a CORE algorithm, which was applied to SDSS SURVEY-PRIMARY point sources with g < 22 or r < 22 (Myers et al. 2015). These point sources were further selected by the XDQSOz algorithm (Bovy et al. 2012), which imposed a probability higher than 20% of being a quasar at redshift z_s > 0.9, with an additional Wide-field Infrared Survey Explorer optical color cut to further reduce stellar contamination. The quasar sample is almost perfectly covered in the sky by the Legacy Surveys, as shown in Figure 1, and we find the overlapped footprint has an effective area of 4699 deg².

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As in Xu et al. (2022b), we employ the z-band 10σ point-source depth to assess the mass completeness of photometric objects. This analysis is conducted within the matched footprint of the Legacy Surveys and the eBOSS quasar sample. As presented in Figure 2, 90% of the regions covered by the Legacy Surveys are deeper than 22.34 in z band. Therefore, we take z = 22.34 as the galaxy depth for the Legacy Surveys.

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We calculate the galaxy physical properties using the SED code CIGALE (Boquien et al. 2019) with grz-band fluxes. We adopt the stellar spectral library provided by Bruzual & Charlot (2003) to build up stellar population synthesis models. The initial stellar mass function in Chabrier (2003) is assumed. We take Z/Z_⊙ = 0.4, 1.0, and 2.5 as three different metallicities in our model. A delayed star formation history $\phi (t)\simeq t\exp (-t/\tau )$ is taken, with the timescale τ varying from 10⁷ to 1.258 × 10¹⁰ yr with an equal logarithm space of 0.1 dex. We apply the starburst reddening law of Calzetti et al. (2000) to consider the dust attenuation, in which the color excess E(B − V) changes from 0 to 0.5.

Following Xu et al. (2022b), we select the deepest 50 deg² regions in the Legacy Surveys, comprising 738 bricks, where the z-band 10σ point-source depth exceeds 23.37, to study the stellar mass completeness C₉₅(M_*), with the photometric redshift (z_p) calculated by Zhou et al. (2021). C₉₅(M_*) is defined such that 95% of the galaxies are brighter than C₉₅(M_*) in z band for a given stellar mass M_*. In Figure 3, we show the stellar mass z-band magnitude distribution of the deep photometric sample at redshift 0.8 < z_p < 1.0. We mark C₉₅(M_*) with the red line in Figure 3, from which the complete stellar mass is 10^10.80 M_⊙ at redshift 0.8 < z_s < 1.0 for the z-band depth of 22.34. Hence, we use the stellar mass of photometric objects at the interval of [10^10.80, 10^11.80] M_⊙ at redshift 0.8 < z_s < 1.0 separated by four equal redshift bins. Furthermore, to constrain the SHMR at lower masses, we incorporate the incomplete stellar mass bins within the range of [10^10.00, 10^10.80] M_⊙ through appropriate modeling, as illustrated later.

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We follow the approach in Xu et al. (2022a) to average the result at different sky regions and redshift bins with proper weights to acquire the final ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$.

For representational simplicity, let ${ \mathcal A }\equiv {\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$. Supposing ${ \mathcal A }$ is calculated for N_r redshift bins and N_s sky regions, we further divide each region into N_sub subregions for error estimation. In our specific scenario, N_r is set to 4, corresponding to the redshift bins [0.80, 0.85], [0.85, 0.90], [0.90, 0.95], and [0.95, 1.00]. Additionally, N_s is defined as 3, representing BASS+MzLS NGC, DECaLS SGC, and DECaLS NGC, while N_sub is established as 10. According to Equation (1), ${{ \mathcal A }}_{i,j,k}$ can be measured in the ith redshift bin, jth sky region, and kth subsample through the Landy–Szalay estimator (Landy & Szalay 1993). First, we average the measurements at different sky regions using the region area w_s,j as a weighting factor to take the sky area difference into account:

$$\begin{eqnarray}&&{{ \mathcal A }}_{i,k}=\displaystyle \frac{{\sum }_{j=1}^{{N}_{{\rm{s}}}}{{ \mathcal A }}_{i,j,k}{w}_{{\rm{s}},{\rm{j}}}}{{\sum }_{j=1}^{{N}_{{\rm{s}}}}{w}_{{\rm{s}},{\rm{j}}}}.\end{eqnarray} \tag{ 2 }$$

Second, we obtain the mean values and the error vector σ_i of the mean values for each redshift bin from N_sub subsamples:

$$\begin{eqnarray}&&{{ \mathcal A }}_{i}={\sum }_{k=1}^{{N}_{\mathrm{sub}}}{{ \mathcal A }}_{i,k}/{N}_{\mathrm{sub}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }_{i}^{2}=\displaystyle \frac{1}{({N}_{\mathrm{sub}}-1){N}_{\mathrm{sub}}}\displaystyle \sum _{k=1}^{{N}_{\mathrm{sub}}}{\left({{ \mathcal A }}_{i,k}({r}_{{\rm{p}}})-{{ \mathcal A }}_{i}({r}_{{\rm{p}}})\right)}^{2}.\end{array}\end{eqnarray} \tag{ 4 }$$

Finally, the results from different redshift bins are averaged according to the error vector σ_i . Let ${w}_{i}({r}_{{\rm{p}}})={\sigma }_{i}^{-1}({r}_{{\rm{p}}})$,

$$\begin{eqnarray}&&{ \mathcal A }=\displaystyle \frac{{\sum }_{i=1}^{{N}_{{\rm{r}}}}{{ \mathcal A }}_{i}{w}_{i}^{2}}{{\sum }_{i=1}^{{N}_{{\rm{r}}}}{w}_{i}^{2}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{1}{{\sum }_{i=1}^{{N}_{{\rm{r}}}}{w}_{i}^{2}}}.\end{eqnarray} \tag{ 6 }$$

We measure ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ and the autocorrelation w_p of quasars in the radial range of 0.1 h⁻¹ Mpc < r_p < 15 h⁻¹ Mpc. These scales are sufficiently broad to capture the distributions of both centrals and satellites. The measurements of ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ are depicted as data points with error bars in Figure 4. The error bars represent the vector σ. These measurements are overall good across all stellar mass bins within the entire radial range, except for the highest-stellar-mass bin [10^11.6, 10^11.8] M_⊙. Additionally, the autocorrelation w_p of quasars, with the weights assigned in the same way as in Hou et al. (2021), is presented in Figure 5 to enhance the precision in constraining the mean host halo mass of quasars.

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We employ the CosmicGrowth simulation suite (Jing 2019), a grid of high-resolution N-body simulations utilizing the computationally efficient adaptive parallel P³M method (Jing & Suto 2002; Xu & Jing 2021). We utilize the ΛCDM simulation with the following cosmological parameters: Ω_m,0 = 0.268, Ω_Λ,0 = 0.732, h = 0.71, n_s = 0.968, and σ₈ = 0.83. This simulation comprises 3072³ dark matter particles within a 600 h⁻¹ Mpc box, with a softening length of η = 0.01 h⁻¹ Mpc. Groups are characterized with the friends-of-friends algorithm (Davis et al. 1985), employing a linking length of 0.2 times the mean particle separation. The halos are then processed with HBT+ (Han et al. 2012, 2018), to identify subhalos and trace their evolution histories. Merger timescales of the subhalos with fewer than 20 particles are estimated using the fitting formula in Jiang et al. (2008). Subhalos that have already merged into central subhalos are disregarded. The adopted simulation has a mass resolution of m_p = 5.54 × 10⁸ h⁻¹ M_⊙, which proves to be sufficiently fine for our study. We utilize snapshot 76 at a redshift of approximately 0.92 to align with the observations.

To associate galaxies with (sub)halos in the N-body simulation, we implement the SHAM method. We utilize the widely used five-parameter formula for the SHMR, a double power law with a constant scatter (Wang & Jing 2010; Yang et al. 2012; Moster et al. 2013; Xu et al. 2023):

$$\begin{eqnarray}&&{M}_{* }=\left[\displaystyle \frac{2k}{{\left({M}_{\mathrm{acc}}/{M}_{0}\right)}^{-\alpha }+{\left({M}_{\mathrm{acc}}/{M}_{0}\right)}^{-\beta }}\right].\end{eqnarray} \tag{ 7 }$$

Here, M_acc is defined as the virial mass M_vir of the (sub)halo when it was last to become a central dominant object, where M_vir is defined through the fitting formula in Bryan & Norman (1998). The scatter in $\mathrm{log}({M}_{* })$ at a given M_acc is assumed to follow a Gaussian distribution with a width of σ. The same set of parameters is applied for both centrals and satellites (Wang & Jing 2010; Behroozi et al. 2019; Xu et al. 2023).

In addition to the standard SHAM parameters described above, given the inclusion of measurements from incomplete stellar mass bins, we introduce four additional parameters k₁, k₂, k₃, and k₄ to account for the incompleteness of the four stellar mass bins. The modeled results from the simulation are obtained by multiplying the ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ from the entire sample by the incompleteness, under the assumption that the incompleteness only affects the amplitude of the observed ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$.

To associate quasars with (sub)halos, we assume that the probability that a (sub)halo hosts a quasar follows a Gaussian distribution of logarithmic halo mass ${\mathrm{log}}_{10}[{M}_{\mathrm{acc}}/({h}^{-1}\,{M}_{\odot })]$, with a mean value μ and a dispersion σ_q. We use the same μ and σ_q for both halos and subhalos, Further, we introduce an additional parameter B to adjust the overall probability for a subhalo to host a quasar relative to a halo at the same mass. This ensures that the subhalos have B times the probability of hosting a quasar relative to a halo of the same mass. Under the above schedule, the satellite fraction f_s of quasars is defined as:

$$\begin{eqnarray}&&{f}_{\mathrm{sate}}=\displaystyle \frac{{{BN}}_{\mathrm{sate}}}{{{BN}}_{\mathrm{sate}}+{N}_{\mathrm{cen}}},\end{eqnarray} \tag{ 8 }$$

where N_sate and N_cen represent the numbers of subhalos and halos selected based on the same Gaussian distribution.

After assigning galaxies and quasars to (sub)halos, we compute the correlation functions using Corrfunc (Sinha & Garrison 2020) in the simulation. To compare with the measurements in observations, we define the χ² as:

$$\begin{eqnarray}\begin{array}{rcl}{\chi }^{2} & = & \displaystyle \sum _{i=1}^{{N}_{{\rm{m}}}}\displaystyle \sum _{j=1}^{{N}_{{\rm{r}}}}(\displaystyle \frac{{\left({{ \mathcal A }}_{i}^{\mathrm{PAC}}({r}_{j})-{{ \mathcal A }}_{i}^{\mathrm{AM}}({r}_{j})\right)}^{2}}{{\sigma }_{i}^{2}({r}_{j})}\\ & & +\,\displaystyle \sum _{j=1}^{{N}_{{\rm{r}}}}\displaystyle \frac{{\left({w}_{{\rm{p}}}({r}_{j})-{w}_{{\rm{p}}}^{\mathrm{AM}}({r}_{j})\right)}^{2}}{{\sigma }^{2}({r}_{j})}.\end{array}\end{eqnarray} \tag{ 9 }$$

Here, N_m and N_r denote the numbers of stellar mass bins and of radial bins, respectively. We explore the parameter space using the Markov Chain Monte Carlo (MCMC) sampler emcee (Foreman-Mackey et al. 2013), employing maximum likelihood analyses for the three sets of parameters {M₀, α, β, k, σ}, {k₁, k₂, k₃, k₄}, and {μ, σ_q, B}.

The best-fit results and errors for the parameters are presented in the first row of Table 1. Additionally, the joint posterior distributions of the parameters are illustrated in Figure A1 using corner (Foreman-Mackey 2016). The best-fit ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ and w_p results from the model are shown in Figures 4 and 5 with solid lines. The fits are generally good for all stellar mass bins, both on the small and large scales.

Table 1. The Best-fit Parameter Results and Errors for the SHMR, Incompleteness, and Quasar–Halo Connection, Assuming a Gaussian Distribution for the Quasar Probability

${\mathrm{log}}_{10}({M}_{0})$	α	β	${\mathrm{log}}_{10}(k)$	σ	μ	σq	k1	k2	k3	k4	B
${11.83}_{-0.07}^{+0.06}$	${0.39}_{-0.07}^{+0.05}$	${2.70}_{-0.42}^{+0.17}$	${10.21}_{-0.06}^{+0.06}$	${0.25}_{-0.05}^{+0.05}$	${13.01}_{-0.13}^{+0.20}$	${0.65}_{-0.12}^{+0.11}$	${0.43}_{-0.05}^{+0.03}$	${0.62}_{-0.05}^{+0.05}$	${0.70}_{-0.05}^{+0.04}$	${0.83}_{-0.05}^{+0.04}$	${1.24}_{-0.22}^{+0.26}$
${11.85}_{-0.05}^{+0.05}$	${0.38}_{-0.05}^{+0.05}$	${2.77}_{-0.33}^{+0.10}$	${10.22}_{-0.04}^{+0.05}$	${0.25}_{-0.04}^{+0.04}$	${12.72}_{-0.14}^{+0.12}$	0.50 (set)	${0.44}_{-0.03}^{+0.02}$	${0.65}_{-0.04}^{+0.03}$	${0.71}_{-0.04}^{+0.05}$	${0.86}_{-0.05}^{+0.05}$	${1.20}_{-0.19}^{+0.24}$

Note. The first row displays the fiducial results with all parameters free, while the second row presents the results for the remaining parameters after setting σ_q to 0.5. M₀ is in units of h⁻¹ M_⊙ and k is in M_⊙.

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For quasars, we find that μ, σ_q, and B are ${13.01}_{-0.13}^{+0.20}$, ${0.65}_{-0.12}^{+0.11}$, and ${1.24}_{-0.22}^{+0.26}$, respectively. The results indicate a very high satellite fraction of quasars, with ${f}_{\mathrm{sate}}={0.29}_{-0.06}^{+0.05}$. B is equal to 1 within the 1σ error, implying that subhalos have nearly the same chance of hosting quasars as the halos and that quasar activity is not affected by the larger halo environment. After determining the host halo masses for subhalos and adjusting the number density of quasars to match the observed value of approximately 1.29 × 10⁻⁵ h³ Mpc⁻³, we illustrate the HOD of our quasar sample in Figure 6. We obtain median host halo masses for central and satellite quasars of ${\mathrm{log}}_{10}[{M}_{{\rm{h}},\mathrm{cen}}^{\mathrm{med}}/({h}^{-1}\,{M}_{\odot })]={12.05}_{-0.60}^{+0.60}$ and ${\mathrm{log}}_{10}[{M}_{{\rm{h}},\mathrm{sat}}^{\mathrm{med}}/({h}^{-1}\,{M}_{\odot })]={12.90}_{-0.72}^{+0.68}$. From the HOD, we observe that although the satellite fraction of quasars looks very high, each massive halo, on average, does not host more than one satellite quasar, due to the low total number density of quasars. To more effectively depict the mass distributions, we also present the host halo mass distributions for central and satellite quasars in Figure B1.

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For galaxies, we present the SHMR at 0.8 < z_s < 1.0 in Figure 7, with ${\mathrm{log}}_{10}[{M}_{0}/({h}^{-1}\,{M}_{\odot })]={11.83}_{-0.07}^{+0.06}$, $\alpha ={0.39}_{-0.07}^{+0.05}$, $\beta ={2.70}_{-0.42}^{+0.17}$, ${\mathrm{log}}_{10}(k/{M}_{\odot })={10.21}_{-0.06}^{+0.06}$, and $\sigma ={0.25}_{-0.05}^{+0.05}$, extending the findings from Xu et al. (2023) to higher redshifts. Our results underscore the potential of leveraging quasars as high-redshift tracers for investigating galaxy formation. However, larger quasar samples and deeper photometric catalogs are required to more precisely constrain the low-mass end (e.g., β in SHMR) and to explore higher redshifts.

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Moreover, we determine the incompleteness of the photometric samples for the four lowest-stellar-mass bins, yielding ${k}_{1}={0.43}_{-0.05}^{+0.03}$, ${k}_{2}={0.62}_{-0.05}^{+0.05}$, ${k}_{3}={0.70}_{-0.05}^{+0.04}$, and ${k}_{4}={0.83}_{-0.05}^{+0.04}$. These results highlight the capability of the PAC method to constrain the stellar mass incompleteness of photometric samples and utilize the incomplete sample to explore lower-mass regions.

The satellite fraction of quasars in our results is ${f}_{\mathrm{sate}}={0.29}_{-0.06}^{+0.05}$. This high satellite fraction suggests that subhalos have nearly the same probability ($B={1.24}_{-0.22}^{+0.26}$) as halos of hosting quasars for the same host (infall) mass and that the large-scale environment has little effect on quasar activity. To verify the robustness of such a high satellite fraction of quasars, we conducted three tests, as follows. Since BASS is a bit shallower in the g and r bands than DECaLs, we measure ${\bar{n}}_{2}{w}_{{\rm{p}}}({r}_{{\rm{p}}})$ in DECaLS and BASS, respectively, and a comparison of the results shows that the measurement is robust (Figure C1).

Initially, we observe a degeneracy between the parameters μ and σ_q in Figure A1. To investigate the impact of this degeneracy on the satellite fraction of quasars, we fix σ_q at 0.50, a value consistent with Richardson et al. (2012), and constrain the remaining parameters through MCMC analysis. The resulting best-fit parameters are presented in the second row of Table 1, and we find that a high satellite fraction is still necessary to align with the observations and that all parameters remain consistent with the previous results within the 1σ interval. For better clarity, we depict the ${\bar{n}}_{2}{w}_{{\rm{p}}}$ results under these best-fit parameters for two stellar mass bins (10^10.5 M_⊙ and 10^10.9 M_⊙) in Figure 8 using solid blue lines. We find that the measurements can also be well fitted by this model. This test confirms that the satellite fraction of quasars remains unaffected by the degeneracy between μ and σ_q.

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Second, to examine whether a low satellite fraction of quasars, as suggested by some previous studies (Richardson et al. 2012; Shen et al. 2013), is consistent with our measurements, we set the B parameter to be 0.16, corresponding to f_sate = 0.05 according to Equation (8), while keeping the remaining parameters unchanged. The simulation results in two stellar mass bins (10^10.5 M_⊙ and 10^10.9 M_⊙) are presented in Figure 8 with solid orange lines. In this case, we observe that the model fails to match the steep increase of ${\bar{n}}_{2}{w}_{{\rm{p}}}$ at small scales (r_p < 1 h⁻¹ Mpc), suggesting that a higher satellite fraction is required to reproduce the observation.

Third, considering our assumption of a Gaussian format for the quasar probability in our modeling, we aim to verify whether this assumption affects the satellite fraction of quasars. Therefore, we replace the Gaussian format with the error function format for both the central and satellite quasars shown, as follows:

$$\begin{eqnarray}&&P({M}_{{\rm{h}}})=\displaystyle \frac{1}{2}\left[1+\mathrm{erf}\left(\displaystyle \frac{{\mathrm{log}}_{10}({M}_{{\rm{h}}})-{\mathrm{log}}_{10}({M}_{\min })}{{\sigma }_{{\mathrm{log}}_{10}({M}_{{\rm{h}}})}}\right)\right],\end{eqnarray} \tag{ 10 }$$

where P(M_h) is the probability of a halo becoming a quasar. ${\mathrm{log}}_{10}({M}_{\min })$ and ${\sigma }_{{\mathrm{log}}_{10}({M}_{{\rm{h}}})}$ are the characteristic mass scale and transition width of a softened step function. The best-fit parameters are presented in Table 2, where we still find a high satellite fraction of quasars, indicated by the parameter $B={1.22}_{-0.18}^{+0.26}$ with ${f}_{\mathrm{sate}}={0.28}_{-0.04}^{+0.05}$. For a clearer illustration, simulation results in two stellar mass bins (10^10.5 M_⊙ and 10^10.9 M_⊙) are shown in Figure 8 with solid green lines. We observe that the measurements can also be well fitted by this model. To compare the difference between the Gaussian format and the error function, we show the HOD from the error function model in Figure 6. We observe that the HODs for satellites are nearly identical for these two models. Although some discrepancy is found for centrals with M_h > 10^13.0 M_⊙, this does not significantly impact f_sate, due to the low number densities of these massive halos. This test confirms that our results are not sensitive to the assumptions in the quasar–halo connection.

Table 2. The Best-fit Parameter Results and Errors for the SHMR, Incompleteness, and Quasar–Halo Connection, Assuming an Error Function for the Quasar Probability

${\mathrm{log}}_{10}({M}_{0})$	α	β	${\mathrm{log}}_{10}(k)$	σ	${\mathrm{log}}_{10}({M}_{\min })$	${\sigma }_{{\mathrm{log}}_{10}({M}_{{\rm{h}}})}$	k1	k2	k3	k4	B
${11.97}_{-0.07}^{+0.09}$	${0.38}_{-0.08}^{+0.06}$	${2.23}_{-0.32}^{+0.36}$	${10.34}_{-0.08}^{+0.10}$	${0.20}_{-0.05}^{+0.04}$	${12.81}_{-0.44}^{+0.23}$	${1.01}_{-0.16}^{+0.11}$	${0.45}_{-0.05}^{+0.03}$	${0.71}_{-0.06}^{+0.05}$	${0.80}_{-0.05}^{+0.06}$	${0.91}_{-0.05}^{+0.05}$	${1.22}_{-0.18}^{+0.26}$

Note. M₀ and ${M}_{\min }$ are in units of h⁻¹ M_⊙ and k is in M_⊙.

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Our higher satellite fraction of quasars than that determined by Shen et al. (2013) may come from two sources. One is the measurements of the cross-correlation between galaxies and quasars. In this paper, we have used galaxies with stellar mass larger than 10^10.0 M_⊙, compared with the LRGs used by Shen et al. (2013). While we find the slopes of the cross-correlation functions within r_p = 1 h⁻¹ Mpc are consistent for massive galaxies (i.e., >10^11.2 M_⊙) between the two works, our measurement for less massive galaxies also requires more satellite quasars around small central galaxies (see the left panel of Figure 8). The other source is different assumptions for the modeling. We use the abundance-matching method, while Shen et al. (2013) used the HOD method. We note that the satellite fraction is quite sensitive to the HOD forms used for the central quasars in Shen et al. (2013). It changes from ${f}_{\mathrm{sat}}={0.068}_{-0.023}^{+0.034}$ when the error function form is used to ${f}_{\mathrm{sat}}={0.099}_{-0.036}^{+0.046}$ when the lognormal form is used instead. Furthermore, they require satellite quasars to exist only in halos with M_h > 10^13.0 h⁻¹ M_⊙ (see their Figure 14). In comparison, we allow many more small halos with M_h < 10^13.0 h⁻¹ M_⊙ (Figure B1) to host satellite quasars, which are required to reproduce the cross-correlation between quasars and galaxies with stellar mass less than 10^11.0 M_⊙ (e.g., the left panel of Figure 8). Therefore, our higher satellite fraction mainly comes from those satellite quasars in halos with M_h < 10^13.0 h⁻¹ M_⊙ that were not probed by Shen et al. (2013). Because central halos as small as M_h > 10^11.5 h⁻¹ M_⊙ can host quasars in the both works (see their Figure 14 and our Figure 6), it is more reasonable that halos with M_h < 10^13.0 h⁻¹ M_⊙ can host satellite quasars, as there are subhalos that are massive enough.