Quasars Have Fewer Close Companions than Normal Galaxies

We use close companions of quasars to trace the merging history of quasar host galaxies. If the AGN activity appears in a certain stage of a galaxy merger, the number of companions around quasars will be different from that of inactive galaxies. Specifically, if two merging galaxies are still distinguishable when the quasar appears, we will see a pair of galaxies in the field; if the quasar emerges when the two progenitor galaxies have already merged into one galaxy, we will see one single quasar host galaxy rather than a pair. We will discuss this point in Section 5.3 in more detail.

We first perform PSF subtraction to suppress the influence of quasar light on detecting their companions. The PSF model is generated by TinyTim (Krist et al. 2011). TinyTim takes the observation time, the position on the CCD chips and the spectrum shape of the source as input parameters. We use a power-law spectrum with a power-law index of −2 (${F}_{\lambda }\propto {\lambda }^{-2}$) as the input spectrum to mimic the typical spectral energy distribution (SED) of a quasar. We use a modeled PSF rather than an empirical PSF because most quasars in our sample come from programs in which the quasars are not the primary science targets, and no separate PSF star observations are available. In addition, the wings of bright quasars can make it difficult to detect projected companions that are several arcseconds away from these quasars, thus a large PSF image is preferred. A modeled PSF can be as large as 20'', which is difficult for an empirical PSF. The size of the PSF models used in this study is 20'' × 20''.

We use GALFIT (Peng et al. 2002) to perform PSF subtraction. Quasar images are fitted by a PSF component plus a Sérsic profile (Sérsic 1963) for the host galaxy. Examples of PSF subtraction can be found in the Appendix. We run SExtractor on the PSF-subtracted images and select all objects with a projected distance to the quasar of 10 kpc < d < 100 kpc as projected quasar companions. We divide the projected distance range 10 kpc < d < 100 kpc into 9 bins, with the ith bin at 10 × i kpc < d < 10 × (i + 1) kpc. The region where d < 10 kpc is severely influenced by the PSF-subtraction residuals for many quasars and is not analyzed.

Companions selected in this way will be inevitably contaminated by foreground and background objects. In the following text, we use "projected companions" to represent all the companions detected (both physical and unphysical). To estimate the numbers of physical companions, a "background density" representing the background/foreground object surface density is calculated for each quasar. We use the entire image to estimate the background object density. Specifically, the surface number density of physical companions of the ith bin is estimated by

$$\begin{eqnarray}&&{\sigma }_{\mathrm{phys}}({d}_{i},{d}_{i+1})=\displaystyle \frac{N({d}_{i},{d}_{i+1})}{A({d}_{i},{d}_{i+1})C({d}_{i},{d}_{i+1})}-\displaystyle \frac{{N}_{\mathrm{bkg}}}{{A}_{\mathrm{bkg}}},\end{eqnarray} \tag{ 1 }$$

where N(d_i, d_i+1) stands for the number of projected companions with 10 × i kpc < d < 10 × (i + 1) kpc, A(d_i, d_i+1) is the corresponding area, and C(d_i, d_i+1) is the completeness of companion detection (see Section 3.2 for details). N_bkg and A_bkg describe the numbers of objects and the area of the whole image. The number of physical companions with 10 × i kpc < d < 10 × j kpc (1 ≤ i < j ≤ 9) is calculated by summing up the number of physical companions in the corresponding bins:

$$\begin{eqnarray}&&{N}_{\mathrm{phys}}({d}_{i},{d}_{j})=\displaystyle \sum _{i\leqslant k\lt j}{\sigma }_{\mathrm{phys}}({d}_{k},{d}_{k+1})A({d}_{k},{d}_{k+1}),\end{eqnarray} \tag{ 2 }$$

and the error of N_phys(d_i, d_j) is estimated assuming a Poisson distribution for N(d_i, d_i+1) and N_bkg in Equation (1).

Unless specified, in the rest of the paper, "number of companions" refers for the estimated number of physical companions and "distance" means projected distance.

Some companions may be missed or misidentified as a result of imperfect PSF subtraction because it can be difficult to distinguish companions from the PSF-subtraction residuals. The probability of a companion to be detected is mainly influenced by the flux contrast between the companion and the PSF-subtraction residual. We find four factors that have a major influence on the completeness: (1) the flux of the quasar, (2) the accuracy of the PSF models, (3) the angular distance from the quasar to the companion, and (4) the flux of the companion. Factors (1) and (2) vary from quasar to quasar, while factors (3) and (4) are determined by the companion itself. Accordingly, we add simulated companions with different flux and distance for each quasar image to estimate the fraction of missed companions. The simulated companions are generated to be point sources. For each quasar image, we simulate three sets of companions, with absolute magnitude M_abs = −19, −20, −21 in the corresponding band, assuming the companions to have the same redshift as the quasar. The completeness is estimated as a function of projected distance to the quasar. We generate 10 simulated companions at randomized positions in each distance bin. The simulated images are analyzed by the companion detection process described above, according to which we calculate the completeness of the companion detection for each quasar. A simulated companion is regarded as "detected" if an object is detected within 02 from the position of the simulated companion.

In the simulation, we do not consider false positives resulting from the PSF-subtraction residual because the probability for a PSF-subtraction residual to appear right at the position of a simulated companion is negligible. The simulation ensures that most companions of interest can be detected. To exclude false positives in the real images, we visually inspect all the images and remove suspicious detections that are likely PSF-subtraction residuals.

Figure 3 shows the average completeness as a function of companion flux and the distance to the quasar from our simulation. The completeness is larger than 90% even for the faintest companion in the smallest distance bin (10 kpc < d < 20 kpc). For companions that are more than 50 kpc away from the quasar (which corresponds to ∼6'' at z = 1.6), the completeness does not change with the distance, which indicates that the influence of the quasar light becomes negligible at larger distances.

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We test the potential influence of using point sources as simulated companions. We run another simulation where the shape of the companions are exponential disks with an effective radius of r_e = 2 kpc, assuming that the companion has a same redshift as the quasar. The difference between the completeness given by the two simulations is less than 1%.

Here we examine the statistics of quasar companions with −20 < M_abs < −19, −21 < M_abs < −20, and M_abs < −21, regardless of the band in which the quasar was observed. We do not perform K-corrections on the magnitudes. Figure 4 shows the average number of companions around quasars in Sample A. The number is negative in some distance ranges due to the subtraction of background object number density. We define companions with a projected distance of 10 kpc < d < 30 kpc to quasars as "close companions" and calculate the average number of close companions $({\overline{N}}_{\mathrm{comp}})$ around quasars (galaxies). The distance range is chosen because companions at 10 kpc < d < 30 kpc are likely involved in a merging event, while companions at larger distances are not necessarily associated with mergers. A similar distance range has been adopted in previous studies on the galaxy merging rates (e.g., Lambas et al. 2003; Ellison et al. 2008; de Ravel et al. 2009; Man et al. 2012). The average numbers of close physical companions are 0.26 ± 0.05, 0.06 ± 0.03, and 0.05 ± 0.03 for −20 < M_abs < −19, −21 < M_abs < −20, and M_abs < −21 companions, and 0.38 ± 0.07 for all physical companions which have M_abs < −19.

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One issue of this analysis is that the "absolute magnitude" we use here is difficult to be translated to physical properties of companions, given that we do not perform K-corrections on these magnitudes. K-corrections require knowledge of the companion SED, while we usually have only one band measurement. This issue will be addressed in Section 4, by introducing a magnitude cut which is associated with galaxy stellar masses.

3.3.1. Ongoing Merging Systems

In Section 3.3, we show the number of companions around quasars as a function of companion flux, where the flux of companions are described by absolute magnitudes without K-corrections. This result is not suitable for studying the redshift evolution of quasar companions or comparing quasars with control sample galaxies, since different bands are used in the analysis of the quasars (all the eight broad bands of ACS/WFC) and the galaxies (F814W only). It is difficult to perform K-corrections in this work, because most of the quasars have only been observed in one band. Therefore, we use an abundance matching technique to convert stellar mass cuts to magnitude cuts in each band, which allows us to set magnitude cuts consistently at any redshift in all the eight bands. In short, for a given stellar mass M_*, we find a magnitude (denoted by m(M_*)) such that that the number of galaxies that are more massive than M_* (denoted by N(M_*)) is the same as the number of galaxies that are brighter than m(M_*). We describe the detailed procedures below.

The analysis is based on the photometric catalog of the UDS field in the 3D-HST project. The UDS photometric catalog is used because it contains the Johnson B, V, R and SDSS i', z' photometry, which can be converted to HST broadband magnitudes and be compared with the quasars. The conversion is done as follows. The SDSS i', z' magnitudes are converted to Johnson I magnitude according to Jordi et al. (2006), then the BVRI magnitudes are converted to ACS/WFC broadband magnitudes according to Sirianni et al. (2005). The conversion from BVRI to F850LP was not provided in Sirianni et al. (2005), so we used SDSS z' as an approximation of the F850LP magnitudes. We estimate the difference between SDSS z' and ACS/WFC F850LP magnitudes using the Exposure Time Calculator for HST ACS/WFC using typical galaxy templates. The difference is smaller than 0.02 mag. We select galaxies in the UDS photometric catalog with USE_PHOT = 1 to construct the galaxy sample for the abundance matching technique. Note that this galaxy sample is different from the control sample; we only use the UDS field in the five 3D-HST fields and do not set any stellar mass cut on this sample.

For a quasar at redshift z_q, the magnitude cut (in an arbitrary band) corresponding to a stellar mass M_* is estimated as follows. First, all objects with photometric redshift (z_phot) that satisfy ${z}_{q}-0.1\lt {z}_{\mathrm{phot}}\lt {z}_{q}+0.1$ in the UDS catalog are selected, constructing a "magnitude-cut-setting galaxy sample." Typically the redshift-matched galaxy sample contains 1000 ∼ 3000 galaxies. The magnitudes of the objects in the magnitude-cut-setting galaxy sample are corrected to z_q according to the photometric redshifts, i.e., given a galaxy with a photometric redshift z_phot and an apparent magnitude m_raw, we calculate the corrected magnitude by

$$\begin{eqnarray}&&m={m}_{\mathrm{raw}}+5\mathrm{log}[{D}_{L}({z}_{q})/{D}_{L}({z}_{\mathrm{phot}})],\end{eqnarray} \tag{ 3 }$$

where D_L(z) is the luminosity distance. We then count the number of objects with stellar masses larger than the given value M_* (referred to as N(M_*)) in the magnitude-cut-setting galaxy sample. The value of N(M_*) varies from several hundred (for M_* = 10⁹M_⊙) to about 20 (for M_* = 10¹¹M_⊙). Finally, we find the magnitude limit m(M_*) so that there are N(M_*) objects that are brighter than m(M_*).

As an example, Figure 5 illustrates the process of estimating m(10¹⁰M_⊙) in F814W band at z = 1.5. The number of objects in the green and the blue shade is equal. By definition, we can estimate the number of quasar companions with stellar masses larger than M_* by counting objects that are brighter than m(M_*).

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For each quasar image, we set up three magnitude limits, m(10⁹M_⊙), m(10¹⁰M_⊙), and m(10¹¹M_⊙). When comparing quasars with the galaxy control samples, we will discuss companions with $m({10}^{9}{M}_{\odot })\gt m\gt m({10}^{10}{M}_{\odot })$ (faint), $m({10}^{10}{M}_{\odot })\gt m\,\gt m({10}^{11}{M}_{\odot })$ (intermediate), and $m\lt m({10}^{11}{M}_{\odot })$ (bright). Correspondingly, we use Sample B throughout Section 4. Since most quasar host galaxies have stellar mass ${M}_{* }\gt {10}^{10}{M}_{\odot }$, "intermediate" companions are mainly associated with major mergers (mass ratio is close to 1), while "faint" companions are mainly related to minor mergers (mass ratio is larger than 3). The case of "bright" companions is more complicated; these systems might be associated with minor mergers where the quasar host galaxy is the less massive progenitor, or major mergers if the quasar host galaxy is as massive as 10¹¹M_⊙. Studies on quasar host galaxies (e.g., Matsuoka et al. 2014; Yue et al. 2018) show that only a small fraction of quasar host galaxies have stellar masses larger than 10¹¹M_⊙, thus most systems with bright companions should be related to minor mergers. We estimate the completeness of companion detection for companions with m(10⁹M_⊙), m(10¹⁰M_⊙), and m(10¹¹M_⊙), using the same method as described in Section 3.2. The result is presented in Figure 6, which shows that the completeness of companion detection is higher than 95% in the simulated images.

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Similar magnitude limits are calculated for galaxies in the control sample using the F814W magnitude. At z = 3, the faintest companions that we consider in this study have magnitude m_F814W(10⁹M_⊙) = 26.3. At this magnitude, about 2% objects in the 3D-HST catalog have signal-to-noise ratios smaller than 3, and the completeness of companion counting will drop toward fainter magnitudes (and thus higher redshift). This is the reason why our sample only contains z < 3 objects.

We calculate the number of faint, intermediate, and bright companions of quasars in Sample B. Figure 7 shows the average number of physical companions, both for quasars and the control sample galaxies. Quasars have fewer intermediate companions at d < 60 kpc. At larger distances, the number of companions around quasars and galaxies are roughly the same. No significant difference can be seen for faint and bright companions between quasars and galaxies.

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For Sample B quasars, the average numbers of "close companions" (companions with a project distance to quasars of 10 kpc < d < 30 kpc) are

$$\begin{eqnarray}&&\begin{array}{l}{\overline{N}}_{\mathrm{comp},{\rm{Q}}}(\mathrm{faint})=0.233\pm 0.043\\ {\overline{N}}_{\mathrm{comp},{\rm{Q}}}(\mathrm{intermediate})=0.004\pm 0.021\\ {\overline{N}}_{\mathrm{comp},{\rm{Q}}}(\mathrm{bright})=0.012\pm 0.018,\end{array}\end{eqnarray} \tag{ 4 }$$

and the comparison galaxy sample has

$$\begin{eqnarray}&&\begin{array}{l}{\overline{N}}_{\mathrm{comp},{\rm{G}}}(\mathrm{faint})=0.178\pm 0.012\\ {\overline{N}}_{\mathrm{comp},{\rm{G}}}(\mathrm{intermediate})=0.087\pm 0.008\\ {\overline{N}}_{\mathrm{comp},{\rm{G}}}(\mathrm{bright})=0.016\pm 0.005.\end{array}\end{eqnarray} \tag{ 5 }$$

Quasars show a 3.7σ deficit of close intermediate companions, and have numbers of faint and bright companions similar to those of normal galaxies.

Figure 8 shows the average number of close companions of quasars and control sample galaxies as a function of redshift. The average number of companions around control sample galaxies increases toward high redshift, both for faint and bright companions. This result is expected since the universe is more crowded at high redshift. Meanwhile, the average number of companions around quasars does not significantly evolve with redshift.

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Figure 9 shows the average number of close companions as a function of quasar luminosity. More luminous quasars have more faint companions, while the number of intermediate companions decreases with quasar luminosity. Given the statistical error, both trends are not significant. The number of bright companions does not show a luminosity dependence.

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We also examine the relationship between average companion number of quasars and other quasar properties, including:

• 1.  
  Broad absorption line (BAL) features. The SDSS quasar catalogs contain BAL flags, while the Véron catalog does not have such information, thus our BAL and non-BAL quasar sample only contain SDSS quasars. We find that BAL quasars and non-BAL quasars have consistent numbers of companions (both faint and intermediate) at a 1σ level.
• 2.  
  Radio loudness. We match our quasar catalog with the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; e.g., Becker et al. 1995; White et al. 1997) survey catalog. We divide the quasars that were covered by the FIRST survey into two subsamples, namely, "radio-loud (RL) quasars" and "radio-quiet (RQ) quasars," according to whether they were detected by the FIRST survey. We find that RL quasars tend to have more "faint" companions than RQ quasars. The average number of "faint" companions is 0.523 ± 0.144 around RL quasars and 0.153 ± 0.047 around RQ quasars, which is a 2.4σ difference. The two samples have similar numbers of intermediate companions (consistent at a 1σ level).
• 3.  
  IR brightness. We match our quasar catalog with the Wide-field Infrared Survey Explorer (WISE; e.g., Wright et al. 2010) ALLWISE source catalog. Since WISE W3 and W4 are usually not deep enough for faint quasars, we use the WISE W1 and W2 fluxes to estimate the rest frame 2μm magnitude (M_2μ) assuming that the near-infrared SED of quasars can be represented by a power law. We then evenly divide the quasars into two subsamples according to their M_2μ − M_i(z = 2) color, referred to as "IR-bright quasars" and "IR-faint quasars," respectively. One potential problem is that the near-infrared SED of some quasars cannot be well fitted by a single power law (e.g., Glikman et al. 2006; Hernán-Caballero et al. 2016). To investigate the influence of fitting quasar SEDs by a single power law, we use the quasar spectrum template in Hernán-Caballero et al. (2016) to fit the WISE W1 and W2 fluxes of the quasars and define the two subsamples based on the template-estimated rest frame 2 μm magnitude. Among all the IR-bright (IR-faint) quasars defined using the power-law fit, 11.6% are classified as IR-faint (IR-bright) in the template-based classification. In both cases, IR-bright and IR-faint quasars have similar numbers of faint and intermediate companions (consistent at a 1.5σ level).

Our sample consists of quasars from the SDSS quasar catalogs and the Véron quasar catalog that have been observed by HST ACS/WFC broadband imaging. The parent sample (SDSS + Véron) includes almost all quasars known to date. Selection effects may be introduced by selecting quasars that have ACS/WFC imaging. Among the 532 quasars in our sample, 402 of them (76%) were observed by ACS/WFC for purposes that were irrelevant to AGN science (e.g., photometric surveys, studies on supernovae or local galaxies). Using the subsample of quasars which were observed in AGN-unrelated programs, we estimate the average companion numbers around quasars to be 0.194 ± 0.052, 0.003 ± 0.028, and 0.005 ± 0.023 for faint, intermediate, and bright companions. These numbers are close to the result we get using the whole sample. For the rest of the sample, the goal of the original HST program was related to AGN science, which could introduce complicated selection effect, e.g., for the programs that targeted a specific class of AGN that could have higher merger fraction.

Our results described in Section 4.2 are based on several assumptions, which may also introduce systematic errors. The main assumption we make is that the companion fluxes can be converted to stellar masses as described in Section 4.1. This cannot be applied to individual objects. However, if quasar companions follow the same stellar mass and flux distribution as normal galaxies, our method will provide the correct numbers of companions that fall in a certain stellar mass range. This method ignores the possible influence of quasars on their companions, which is a complicated effect and is difficult to correct. When applying the results, this potential systematic error must be kept in mind. On the other hand, the "primary results" in Section 3.3 are free from this systematic error.

Our results suggest that there is a deficit of companions around quasars compared with inactive galaxies. We interpret the difference as a result of the difference between the merging history of quasars (especially merger-triggered ones) and normal galaxies. Here we first review the merger-triggering model of quasars briefly and discuss how many companions would we expect around a merger-triggered quasar.

According to simulations (e.g., Lotz et al. 2010), the evolution of a galaxy merger will experience five stages: the first encounter, the largest separation, the second encounter, the final coalesce, and the post-merger remnant. Each stage lasts for ≲0.5 Gyr. The typical lifetime of a quasar is 10 ∼ 100 Myr (e.g., Hopkins et al. 2008), which means that the morphological properties of quasar host galaxies will not change significantly during the life of the quasar.

Previous simulations on galaxy major mergers triggering quasars (e.g., Di Matteo et al. 2005; Springel et al. 2005; Hopkins et al. 2006; Newton & Kay 2013) have suggested that quasars emerge at the final coalesce stage. This can be understood in the following picture. In the merger-triggering model, the quasar activity emerges when strong gas inflows feed the SMBH. This process requires the gas content to be highly disturbed. Simulations of galaxy major mergers show that the disturbed features are the most prominent at the second encounter stage. It takes some time for the gas to reach and feed the SMBH, thus strong quasar activities are expected to emerge at the final coalesce. As an observational evidence, Ellison et al. (2013) found that post-mergers in SDSS have a high AGN fraction.

With some simple calculations, we derive the fraction of merger-triggered quasars using the number of close companions. Assuming that quasars are either triggered by secular evolution or mergers, and the fraction of merger-triggered quasar is α. We use ${\overline{N}}_{\mathrm{comp},\mathrm{QS}}$ and ${\overline{N}}_{\mathrm{comp},\mathrm{QM}}$ to denote their average number of companions, where "Q" means "quasar," "S" stands for "secular evolution" and "M" means "merger." The observed average companion number of quasars should be ${\overline{N}}_{\mathrm{comp},{\rm{Q}}}=(1-\alpha ){\overline{N}}_{\mathrm{comp},\mathrm{QS}}+\alpha {\overline{N}}_{\mathrm{comp},\mathrm{QM}}$, from which we have

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{\overline{N}}_{\mathrm{comp},\mathrm{QS}}-{\overline{N}}_{\mathrm{comp},{\rm{Q}}}}{{\overline{N}}_{\mathrm{comp},\mathrm{QS}}-{\overline{N}}_{\mathrm{comp},\mathrm{QM}}}.\end{eqnarray} \tag{ 6 }$$

We further assume that (1) secular-evolution-triggered quasars have the same number of companions as normal galaxies (${\overline{N}}_{\mathrm{comp},\mathrm{QS}}\approx {\overline{N}}_{\mathrm{comp},{\rm{G}}}$, where "G" means inactive galaxies), since the secular evolution will not influence the merging process; and (2) merger-triggered quasars have no close companions (${\overline{N}}_{\mathrm{comp},\mathrm{QM}}\approx 0$), since the two progenitor galaxies of the merger-triggered quasar have already merged into a single galaxy. We then have

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{\overline{N}}_{\mathrm{comp},{\rm{G}}}-{\overline{N}}_{\mathrm{comp},{\rm{Q}}}}{{\overline{N}}_{\mathrm{comp},{\rm{G}}}}.\end{eqnarray} \tag{ 7 }$$

Figure 7 indicates that the difference between the average number of companions of quasars and galaxies varies with the companion stellar mass. As a result, α depends on the companion mass. This result is not surprising since we expect that major mergers and minor mergers have different probabilities to trigger quasars. We can interpret α(M_companion) as the fraction of quasars that are triggered by a merger where one progenitor galaxy had a stellar mass of M_companion. Equations (4) and (5) describe the average number of companions of our sample. According to Equation (7), we have α(faint) = −0.31 ± 0.26, α(intermediate) = 0.95 ± 0.25 and α(bright) = 0.23 ± 1.14. Since intermediate and faint companions are associated with major and minor mergers, respectively, our toy model indicates that there should be a significant fraction of quasars that are triggered by major mergers, and that minor mergers have a small contribution in quasar triggering. The large error for the bright companions does not allow for a strong conclusion.

We now discuss the impact of our assumptions in the analysis above. The first assumption is that secular-evolution-triggered quasars have the same number of companions as normal galaxies. Close neighbors of galaxies can disturb their gas kinematics and lead to gas inflows. However, gas inflows generated in this way are usually not strong enough to feed a quasar, thus the environmental influence should be minor. Previous studies have also suggested that secular evolution is only responsible for faint AGNs (e.g., Treister et al. 2012). Moreover, according to Equation (6), a larger number of close companions around secular-evolution-triggered quasars will lead to a higher fraction of merger-triggered quasars. The second assumption is that there are no companions around merger-triggered quasars. This assumption is clearly oversimplified. Our main idea is that, in the picture of a merger-triggered quasar discussed above, the quasar host galaxy is a galaxy merger that has entered the final coalesce stage, and we can only see one galaxy rather than a pair. Two possible errors come into this assumption. On the one hand, a merger-triggered quasar may not show up exactly in the final coalesce stage. It is possible that the quasar is triggered earlier when the two merging galaxies are still distinguishable. We suggest that the influence of this possible error should be small, however, because we only count companions with distance larger than 10 kpc, and most observed merging galaxy pairs are closer. On the other hand, if there are nearby galaxies that are not involved in this merging event, we will have a non-zero companion number for merger-triggered quasars. Given the distance cut we applied (10 kpc < d < 30 kpc), such cases should be rare. In both cases, Equation (6) suggests that, if ${\bar{N}}_{\mathrm{comp},\mathrm{QM}}\gt 0$, the fraction of merger-triggered quasars will become even larger, which will further strengthen our conclusion.

Conclusions from previous studies on AGN merger fractions have been ambiguous. There are both results indicating enhanced merging fractions of AGN (e.g., Ellison et al. 2011; Silverman et al. 2011; Satyapal et al. 2014; Fan et al. 2016; Goulding et al. 2018; Weston et al. 2017) and indicating no difference between AGN and normal galaxies (e.g., Cisternas et al. 2011; Schawinski et al. 2011; Kocevski et al. 2012; Villforth et al. 2017). Our result suggests that there should be a significant fraction of major-merger-triggered quasars. We consider several possible reasons for this ambiguity.

First, most of these studies used distinct AGN samples. The AGN samples vary from emission-line-ratio selected (e.g., Ellison et al. 2011), near-IR selected (e.g., Satyapal et al. 2014; Fan et al. 2016; Goulding et al. 2018; Weston et al. 2017), X-ray selected (e.g., Cisternas et al. 2011; Silverman et al. 2011; Kocevski et al. 2012; Villforth et al. 2017), to optical selected (this work). We notice that all the studies using near-IR-selected AGN samples reach a conclusion that AGN have an enhanced merging fraction, while most of the X-ray selected AGN samples do not show a significant difference from inactive galaxies. This result can be explained if different populations of AGN emerge in different stages of galaxy mergers. The luminosities of the AGN samples are also different, which is believed to have a crucial influence on AGN merging fraction. Previous observations using optical data focused mainly on low-luminosity AGNs to avoid the strong emission from the central nuclei. Most studies on high-luminosity AGNs are based on either X-ray or infrared data (for obscured ones). Comparing to these studies, our sample consists of optical-selected AGNs and spans a wide range of luminosity $(-31\lt {M}_{i}(z=2)\lt -23)$.

Second, the method of these studies might introduce some biases. Most of the previous studies used disturbed features in quasar host galaxies as indicators of recent merger events. There have been arguments that these features should be able to survive until the quasar activity emerges (e.g., Cisternas et al. 2011; Villforth et al. 2017). However, this depends on the model of galaxy mergers and AGN triggering, which is highly uncertain and varies from object to object. Even if the average timescale of disturbed features may be long enough, it is still possible to miss some mergers and thus underestimate the merger fraction of AGN. The uncertainty in identifying disturbed features in AGN host galaxies might be more severe for bright unobscured AGNs that need PSF subtraction, given the difficulty of PSF modeling.

It is also difficult to distinguish major mergers and minor mergers based on the host galaxy morphology. If minor mergers do not contribute to the triggering of quasars (as indicated by our result), including them as "recent merging systems" will increase the number of identified merging systems and introduce extra uncertainties when testing the major-merger-triggering mechanism.

In comparison, we identify mergers by counting companions. This will not introduce significant biases against the late-stage mergers, where the disturbed features in the galaxies might have already faded away. It is also more straightforward to distinguish major and minor mergers since we can estimate the companion mass. Counting companions is an accessible way for nearly all populations of AGN, and does not require superb angular resolution, which makes it easier to build a large, unbiased sample.

Keeping the possible biases in mind, we find that most of the results are consistent with the picture where:

(1) Mergers are the main triggering mechanism for high-luminosity AGNs, and secular evolution is mainly responsible for low-luminosity AGNs. There have been simulations claiming that secular evolution is not powerful enough to trigger the most luminous quasars (e.g., Treister et al. 2012). Previous observations have also reported that the merger fraction increases with AGN luminosity (e.g., Fan et al. 2016). We find that the average number of intermediate companions decreases with quasar luminosity. According to Figure 9 and Equation (7), our result indicates that luminous quasars are more likely to be triggered by major mergers.

(2) Merger-triggered AGNs evolve from an obscured to an unobscured phase. The transition happens when the radiation and material outflows blow away the dust around the active nucleus. Consequently, IR-selected AGNs (which are dustier) might represent an earlier stage of AGN evolution, and it is easier to detect the disturbed features in their host galaxies than other populations of AGN. This picture is supported by simulations (e.g., Di Matteo et al. 2005; Hopkins et al. 2008), and can explain the discrepancy with previous observations, which reported that IR-selected AGNs have a larger merging fraction than X-ray-selected AGNs.

5.3.1. Merger Rate of AGN versus AGN Rate in Mergers

In comparison to works such as ours, which measures the fraction of mergers in quasars, some studies instead compared the fraction of AGN in merging and non-merging systems, quantified by the "AGN fraction ratio":

$$\begin{eqnarray}&&R=\displaystyle \frac{P(\mathrm{AGN}| \mathrm{merger})}{P(\mathrm{AGN}| {\rm{non \mbox{-} merger}})},\end{eqnarray} \tag{ 8 }$$

where $P(\mathrm{AGN}| \mathrm{merger})$ ($P(\mathrm{AGN}| {\rm{non \mbox{-} merger}})$) is the probability of finding an AGN in a (non-)merging system. Previous studies found that merging systems are more likely to host AGN (e.g., R ∼ 2−7 in Goulding et al. 2018, R ∼ 5−17 in Weston et al. 2017), and concluded that galaxy mergers are the dominant triggering mechanism of the AGN in their sample. Using Bayesian analysis, we can calculate the AGN fraction ratio using the merger ratio in AGN, and thus compare our result directly with previous studies. According to Bayes' theorem,

$$\begin{eqnarray}\begin{array}{rcl}P(\mathrm{AGN}| \mathrm{merger}) & = & \displaystyle \frac{P(\mathrm{merger}| \mathrm{AGN})P(\mathrm{AGN})}{P(\mathrm{merger})}\\ P(\mathrm{AGN}| {\rm{non \mbox{-} merger}}) & = & \displaystyle \frac{P({\rm{non \mbox{-} merger}}| \mathrm{AGN})P(\mathrm{AGN})}{P({\rm{non \mbox{-} merger}})},\end{array}\end{eqnarray} \tag{ 9 }$$

thus

$$\begin{eqnarray}\begin{array}{rcl}R & = & \displaystyle \frac{P({\rm{non \mbox{-} merger}})}{P(\mathrm{merger})}\times \displaystyle \frac{P(\mathrm{merger}| \mathrm{AGN})}{P({\rm{non \mbox{-} merger}}| \mathrm{AGN})}\\ & = & \displaystyle \frac{1-P(\mathrm{merger})}{P(\mathrm{merger})}\times \displaystyle \frac{P(\mathrm{merger}| \mathrm{AGN})}{1-P(\mathrm{merger}| \mathrm{AGN})},\end{array}\end{eqnarray} \tag{ 10 }$$

where we apply P(merger) + P(non-merger) = 1.

In our sample, the estimated major-merger-triggered quasar fraction is α = 0.95 ± 0.25, with a 3σ lower limit of 0.22. We assume that the secular-evolution-triggered quasars have the same merger fraction as inactive galaxies (i.e., ${P}_{{\rm{S}}}(\mathrm{merger}| \mathrm{AGN})\,=P(\mathrm{merger})$). We also have ${P}_{{\rm{M}}}(\mathrm{merger}| \mathrm{AGN})=1$ by definition. We adopt the merger fraction of inactive galaxies from Villforth et al. (2017), which gave P(merger) ≈ 0.2. This gives

$$\begin{eqnarray}&&\begin{array}{l}P(\mathrm{merger}| \mathrm{AGN})=\alpha {P}_{{\rm{M}}}(\mathrm{merger}| \mathrm{AGN})\\ \qquad +\ (1-\alpha ){P}_{{\rm{S}}}(\mathrm{merger}| \mathrm{AGN})\geqslant 0.37,\end{array}\end{eqnarray} \tag{ 11 }$$

and R ≥ 2.4 according to Equation (10), which is consistent with previous studies.