A Mock Catalog of Gravitationally-lensed Quasars for the LSST Survey

We use the CosmoDC2 mock galaxy catalog (Korytov et al. 2019) to model the deflector galaxies. CosmoDC2 is a synthetic galaxy catalog that covers an area of 440 deg² out to a redshift of z = 3. CosmoDC2 first identifies dark matter halos in the N − body simulation Outer Rim (Heitmann et al. 2019), and assigns simulated galaxies from the Universe Machine (Behroozi et al. 2019) to the dark matter halos. The algorithm is tuned to match the observed stellar mass versus star-formation rate (M_*, SFR) distributions of galaxies. CosmoDC2 then assigns various of properties to the mock galaxies, including stellar masses, optical-to-near-infrared SEDs, half-light radii and ellipticities, using empirical relations or simulated galaxy spectra library. Korytov et al. (2019) shows that the mock catalog reproduces the observed colors and number counts of galaxies at a wide range of redshifts. The large area of CosmoDC2 catalog ensures a small statistical error of the simulated lens sample, and the galaxy properties provides all the information needed to generate mock observations of lensing systems.

We use singular isothermal ellipsoids (SIE; e.g., Kormann et al. 1994) to model the lensing potential of the deflector galaxies. An SIE lens is parameterized by its ellipticity, e = 1 − q where q is the axis ratio ⁴ , and its Einstein radius,

$$\begin{eqnarray}&&{\theta }_{E}=4\pi {\left(\displaystyle \frac{\sigma }{c}\right)}^{2}\displaystyle \frac{{D}_{\mathrm{ds}}}{{D}_{{\rm{s}}}}\end{eqnarray} \tag{ 1 }$$

where σ is the line-of-sight velocity dispersion of the galaxy, D_s is the angular diameter distance from the observer to the source, and D_ds is the angular diameter distance from the deflector to the source.

We estimate the velocity dispersion of a galaxy using its stellar mass. Specifically, we convert stellar masses to dynamical masses using a stellar-to-dynamical mass ratio, and calculate the velocity dispersion by applying the virial theorem (e.g., Cappellari et al. 2006):

$$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{{GM}_{{\rm{d}}{\rm{y}}{\rm{n}}}}{{KR}_{e}}}.\end{eqnarray} \tag{ 2 }$$

We adopt M_*/M_dyn = 0.557, which gives a good description for galaxies in a wide mass (100 km s⁻¹ < σ < 300 km s⁻¹) and redshift (0 < z < 1) range (e.g., Bezanson et al. 2011, 2012). Since early-type galaxies dominate the deflector population in galaxy-scale lenses (e.g., Oguri 2006; Inada et al. 2012), we adopt the typical scaling factor K = 6 for early-type galaxies (e.g., Beifiori et al. 2014; Nigoche-Netro et al. 2016). We do not use the Sérsic-index-dependent scaling factor as suggested in, e.g., Bertin et al. (2002), as CosmoDC2 uses a bulge + disk composition to describe the galaxy morphology instead of a Sérsic profile.

Figure 2 compares the derived VDF of the CosmoDC2 mock galaxy catalog with the observed galaxy VDF. The local VDF has been accurately measured spectroscopically (e.g., Choi et al. 2007). We find that the VDF of the CosmoDC2 catalog at z < 0.1 is fully consistent with the recent spectroscopic measurements (Sohn et al. 2017; Hasan & Crocker 2019). At higher redshifts, it is challenging to measure the velocity dispersion of a large galaxy sample and thus the VDF spectroscopically. The right panel of Figure 2 compares the CosmoDC2 catalog VDF with the observed VDF at z < 1.5 from Bezanson et al. (2012), who measure the dynamical mass of galaxies in the COSMOS and UDF fields using photometric data. The CosmoDC2 catalog is also in good agreement with the observed values except at the low-mass end at z > 0.6, which can be due to incompleteness of the galaxy sample in the observations. Figure 2 suggests that our method successfully reproduces the observed VDF at least at z < 1.5.

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In this work, we only include galaxies with σ > 50 km s⁻¹, which corresponds to an Einstein radius limit of ${\theta }_{E,\mathrm{lim}}=4\pi {({\sigma }_{\mathrm{lim}}/c)}^{2}\,=\,0\buildrel{\prime\prime}\over{.} 072$. Lensing systems with smaller θ_E are difficult to identify even with space telescopes like the Hubble Space Telescope (HST).

We notice that the observed M_*–σ relation have significant scatters, which can be partly explained by the effect of galaxy sizes (R_e ). The observed scatter of Equation (2) is ∼0.02dex for the velocity dispersion (e.g., Auger et al. 2010). This small scatter have little influence on the galaxy VDF, and is not considered in this work.

In addition to the deflector galaxy, we add an external shear and convergence to each lensing system. We use the CosmoDC2 catalog to estimate the redshift evolution of external shears and convergences. Specifically, we divide the mock galaxy catalog into redshift bins with Δz = 0.01, then fit the distribution of external convergences (κ) and the two components of the external shear (γ₁, γ₂) in each redshift bin as normal distributions centered at zero. For both κ and γ, the best-fit standard deviation can be well-described by a linear function of $\mathrm{log}(1+z)$:

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{\sigma }_{\kappa } & = & 0.057\times {\rm{l}}{\rm{o}}{\rm{g}}(1+z)-0.001\\ {\sigma }_{({\gamma }_{1}{\rm{o}}{\rm{r}}{\gamma }_{2})} & = & 0.040\times {\rm{l}}{\rm{o}}{\rm{g}}(1+z)-\mathrm{0.001.}\end{array}\end{array}\end{eqnarray} \tag{ 3 }$$

For each lensing system, we use the best-fit normal distribution to generate a random set of (κ, γ₁, γ₂) values. ⁵

Note that the convergence and shears in the CosmoDC2 catalog have some limitations. Specifically, the convergence and shears from ray-tracing simulations depend on the resolution (i.e., the smoothing scale) of the density field, especially at small scales (e.g., Hilbert et al. 2009). To perform the ray-tracing simulation, CosmoDC2 divides the mass particles from the cosmological simulation into discrete shells and estimates the surface mass density of each shell on a HEALPix grid with Nside = 4096, which corresponds to a resolution of 05 on the sky. Korytov et al. (2019) compares the cosmic shear power spectrum of the CosmoDC2 catalog and theoretical predictions, showing that the difference is negligible at ℓ < 1000 and is ≲ 10% at ℓ ∼ 4000. As the values of κ and γ are small (Equation (3)), this systematic uncertainty has effectively no impact on the number of lensed quasars.

We simulate quasars at z < 7.5 using python code SIMQSO ⁶ (McGreer et al. 2013). SIMQSO generates mock catalogs of quasars by randomly sampling the absolute magnitude and the redshift of quasars according to the input QLF. For each simulated quasar, SIMQSO generates a mock spectrum which includes a broken power-law continuum and a number of emission line components, and sets a random sightline to model the intergalactic medium (IGM) absorption based on observed IGM models. We use the default continuum, emission line and IGM parameters in SIMQSO which has been shown to successfully reproduce the observed quasar color distribution (e.g., Ross et al. 2013) and has been widely used to estimate the completeness of quasar surveys (e.g., Jiang et al. 2016; Yang et al. 2016; McGreer et al. 2018; Wang et al. 2019).

We use a double power law to describe the QLF:

$$\begin{eqnarray}&&{\rm{\Phi }}(M,z)=\displaystyle \frac{{{\rm{\Phi }}}_{* }}{{10}^{0.4(M-{M}^{* })(\alpha +1)}+{10}^{0.4(M-{M}^{* })(\beta +1)}}\end{eqnarray} \tag{ 4 }$$

where α and β are faint and bright end slopes of the QLF, M* is the break absolute magnitude and Φ_* is the normalization. Specifically, the QLF we adopt at different redshifts are as follows:

• 1.  
  At 0 < z < 3.5, we adopt the pure luminosity evolution (PLE; for z < 2.2) and the luminosity evolution and density evolution (LEDE; for 2.2 < z < 3.5) model from Ross et al. (2013). We use the relations in the Appendix B of Ross et al. (2013) to convert the absolute magnitudes to M₁₄₅₀, i.e., the absolute magnitude at rest frame 1450 Å.
• 2.  
  At 3.5 < z < 7.5, we compile a number of QLFs at different redshifts from the literature. These QLFs are summarized in Table 1. At 3.5 < z < 6.7, we use linear interpolation to obtain the QLF parameters at a certain redshift. At z > 6.7, we keep α, β and M* fixed to the z = 6.7 value, and apply a density evolution of $\mathrm{log}{{\rm{\Phi }}}_{* }(z)=\mathrm{log}{{\rm{\Phi }}}_{* }(z=6.7)-0.78\times (z-6.7)$, following Wang et al. (2019).

Table 1. QLF Parameters from the Literature

Redshift	$\mathrm{log}{{\rm{\Phi }}}_{* }$	M*	α	β
	(Mpc−3mag−1)	(mag)
0.1 < z < 2.2 a	$-{5.96}_{-0.06}^{+0.02}$	$-{21.364}_{-0.11}^{+0.05}-2.5\times ({1.241}_{-0.028}^{+0.010}z-{0.249}_{-0.017}^{+0.006}{z}^{2})$	$-{1.16}_{-0.04}^{+0.02}$	$-{3.37}_{-0.05}^{+0.03}$
2.2 < z < 3.5 b	$-{5.83}_{-0.25}^{+0.15}-{0.689}_{-0.027}^{+0.021}\times (z-2.2)$	$-{25.00}_{-0.24}^{+0.34}-{0.809}_{-0.166}^{+0.033}\times (z-2.2)$	$-{1.31}_{-0.19}^{+0.52}$	$-{3.45}_{-0.21}^{+0.35}$
z = 3.9 c	−6.58 ± 0.01	−25.36 ± 0.13	−1.30 ± 0.05	−3.11 ± 0.07
z = 5.0 d	$-{7.36}_{-0.81}^{+0.56}$	$-{25.78}_{-1.10}^{+1.35}$	$-{1.21}_{-0.64}^{+1.36}$	$-{3.44}_{-0.84}^{+0.66}$
z = 6.0 e	$-{7.96}_{-0.42}^{+0.28}$	$-{24.90}_{-0.90}^{+0.75}$	$-{1.23}_{-0.34}^{+0.44}$	$-{2.73}_{-0.31}^{+0.23}$
z = 6.7 f	−8.43 ± 0.23	[−24.90]	[−1.23]	−2.51 ± 0.39

Notes. The parameters are defined in Equation (4). Absolute magnitudes in this table are M₁₄₅₀. The square brakets marks quantities that are fixed when fitting the QLFs. Specifically, Wang et al. (2019) only measures the bright end of z ∼ 6.7 QLF, and we fit the double power law by fixing the M* and α values to that of Matsuoka et al. (2018). For 3.5 < z < 6.7, we perform linear interpolation to obtain QLF model parameters. For z > 6.7, we assume a pure density evolution of $\mathrm{log}{{\rm{\Phi }}}_{* }(z)=\mathrm{log}{{\rm{\Phi }}}_{* }(z=6.7)-0.78\times (z-6.7)$, following Wang et al. (2019).

^a The PLE model in Ross et al. (2013). ^b The LEDE model in Ross et al. (2013). ^c The parameterized QLF at z = 3.9 from Akiyama et al. (2018). ^d The parameterized QLF at z = 5 from Kim et al. (2020). ^e The parameterized QLF at z = 6 from Matsuoka et al. (2018; the standard model in their Table 5). ^f The QLF at z ∼ 6.7 by fitting the binned QLF from Wang et al. (2019), with the faint-end slope and M* fixed to the values in Matsuoka et al. (2018).

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Figure 3 shows the QLF used in this work at different redshifts. Both Table 1 and the right panel of Figure 3 suggest a subtle redshift evolution of M*, α and β at high redshift. In addition, a linear interpolation of $\mathrm{log}{{\rm{\Phi }}}_{* }$ is equivalent to an exponential redshift evolution of Φ_*, which has been shown to well describe the number density of high-redshift quasars (e.g., Jiang et al. 2016; McGreer et al. 2018; Wang et al. 2019). Using linear interpolations between redshifts is thus sufficient to accurately capture the evolution of QLFs at high redshift.

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Following the Active Galactic Nuclei (AGN) chapter in the LSST science book (LSST Science Collaboration et al. 2009), we only include quasars with M_i < − 20 in our simulated quasar sample, which is equivalent to M₁₄₅₀ < − 19.1 according to the relation in Ross et al. (2013). For faint AGN below this flux limit, host galaxies start to dominate their emissions. These objects require a survey strategy that is different from classical quasars and is out of the scope of this paper.

The area of the CosmoDC2 catalog is 440 deg², which is 1.067% of the whole sky. To make the simulated lens catalog equivalent to an all-sky catalog, we scale the QLF by a factor of 2%/1.067% = 1.874, so that the number of expected lensed quasars in the CosmoDC2 field equals to 1/50 of the sky. By running 50 random realizations (see Section 2.3), we can obtain a mock catalog that is equivalent to the all-sky area.

After producing the mock deflector galaxies and the simulated background quasars, we generate one realization of the mock lens catalog as follows. We first assign random positions in the CosmoDC2 field to the simulated quasars, and assign random position angles to the mock galaxies. Then, for each galaxy, we identify all quasars that have distances to the galaxy smaller than $5{\theta }_{E,\max }$, where ${\theta }_{E,\max }=4\pi {(\sigma /c)}^{2}$ is the maximum possible Einstein radius for the deflector galaxy. For each matched galaxy-quasar pair, we assign a random external convergence and a random external shear according to the best-fit probabilistic distribution described in Section 2.1, and use GLAFIC (Oguri 2010) to model the lensing configuration. GLAFIC is a software that provides fast and flexible lensing analysis for a variety of deflector mass and source emission models. We then select multiply-imaged quasars into the mock lens catalog.

We run 50 random realizations to generate a mock lens catalog that is equivalent to the all-sky area. For quasars at z > 5, due to their low number density, we run 1000 realizations (i.e., 20 × all-sky area) to reduce the Poisson noise.

We calculate the magnitdues of the simulated lensed quasars in the LSST ugrizy, UKIDSS JHK (Lawrence et al. 2007), and WISE W1, W2 bands. We use the simulated spectra generated by SIMQSO to calculate the synthetic magnitudes of quasars. For deflector galaxies, CosmoDC2 provides the synthetic LSST ugrizy magnitudes and 30 top-hat SED points in the wavelength range 1000Å < λ < 20000Å. We directly adopt the synthetic ugrizy magnitudes from the CosmoDC2 catalog, and estimate the JHK and W1, W2 magnitudes using the SED. Specifically, the SED covers J and H at all redshifts, and we linearly interpolate the SED points to build a spectra and calculate the magnitudes. At low redshifts, the SED does not cover the wavelengths of K, W1 and/or W2. In this case, we fit the SED using the galaxy templates from Brown et al. (2014), and use the best-fit template to estimate the magnitude in that filter.

To further explore how survey strategies work for real data, we generate simulated LSST images in ugrizy bands for the mock lensed quasars. We use point sources to describe the lensed images of the background quasar, and use a bulge (de Vaucouleurs) + disk (exponential) two-component model to describe the foreground galaxy. CosmoDC2 provides the radius and the SED of both components for each galaxy. We then generate mock images using galfit (Peng et al. 2002) for each lensing system. Specifically, the images are created as follows:

• 1.  
  The point-spread function (PSF): the PSF of each image is modeled as an 2D Gaussian profile. The full-width half maximum (FWHM) is drawn from a normal distribution with a standard deviation σ_FWHM = 0.02 × mean, where mean is the mean PSF size of the corresponding filter. The axis ratio is uniformly drawn in [0.95, 1], and the position angle is uniformly drawn from [0, 2π). This approach mimics the PSF variation in real observations.
• 2.  
  Image noises: we follow the methods in the LSST review paper (Ivezić et al. 2019; also see https://smtn-002.lsst.io/) to calculate the errors of image pixels. The random noise of an pixel consists of the background noise and the Poisson noise of the source flux. These noises are calculated based on Equation (5) in Ivezić et al. (2019).

Table 2 summarizes the mean PSF sizes and depths we used in the simulation. Figure 4 shows some examples of the SED and simulated LSST images of a mock lensed quasar, which exhibit a large diversity in observational features.

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The final data products are a mock catalog of lensed quasars and mock LSST images for each simulated lensed quasar. We provide a number of properties in the mock catalog, including the lensing configuration, the time delay of each lensed image, and the magnitudes of the quasar and the deflector. These properties are summarized in Table 3.

The mock catalog in OM10 and this work have a number of significant differences: (1) the two mock catalogs adopt different QLFs and VDFs; (2) we only include quasars with M_i < − 20, while OM10 extend the quasar sample to as faint as M_i ∼ − 18; and (3) OM10 only include lensing systems with separation Δθ > 05, while our mock catalog have more compact lenses down to Δθ ∼ 01.

To perform meaningful comparisons, we select subsamples of the two mock catalogs that have M₁₄₅₀ < − 24, z_qso < 3 and Δθ > 05. In this redshift and magnitude range, the QLFs used in OM10 and this work are close to each other (Figure 7, left panel). We also compare the VDFs adopted in OM10 and this work in the middle panel of Figure 7. In short, OM10 adopt the local VDF from Choi et al. (2007) and assume no redshift evolution. This VDF is close to the CosmoDC2 catalog VDF at z < 0.1, but it predicts higher number density than the CosmoDC2 VDF beyond the local universe. These differences can fully account for the predicted numbers of lensed quasars, shown in the right panel of Figure 7. After applying the luminosity, redshift and lensing separation cuts mentioned above, our mock catalog gives a slightly smaller number of lensed quasars compared to OM10. The main difference between the two mock catalogs appears at z_qso ∼ 2.2, where the OM10 catalog has ∼ 30% more lenses than this work. This can be explained by the difference in the VDFs at z_d ≳ 0.3, where the OM10 VDF is ∼0.3 dex higher than the CosmoDC2 VDF.

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The comparison suggests that our method of generating mock catalog is reliable, and illustrates the importance of using accurate and updated QLFs and VDFs. Another major difference between OM10 and this work is that we apply an absolute magnitude cut of M_i < − 20 for quasars. This cut is used in the LSST science book, and the main motivation is that fainter AGNs have distinct observational features compared to normal type-I quasars. For these objects, host galaxy emissions start to dominate the SEDs and make their morphology extended. Modeling the host galaxy emission is out of the scope of this work, and we expect that these faint AGNs require a very different candidate selection strategy.

AGN variability offers a possible way of finding host-dominated lensed AGNs (e.g., Kochanek et al. 2006). OM10 presented the number of lensed quasars that are brighter than the 10σ limits of yearly stacked images for each sky survey, which can be identified via their variability. Variability analysis will be especially useful in the LSST survey, providing a critical supplement to image-based candidate selection methods.

One important application of lensed quasars is to measure key cosmological parameters, especially the Hubble constant H₀. This relies on measuring the time delays between the lensed images of the background quasar. Compared to doubly imaged ones, quadruply-lensed quasars are preferred since they provide more time delays and give stronger constraints.

Table 4 gives the number of discoverable quadruply-lensed quasars in various sky surveys. Our mock catalog suggests that there are ∼200 quadruply-lensed quasars discoverable in LSST. The impact of survey depth on the number of discoverable lenses is less significant for quads compared to doubly-lensed quasars, which is a result of magnification bias and the criteria of discoverable lenses (Section 3). In general, the fainter image in doubly-lensed systems has a low magnification and is often demagnified $(\overline{\mu }=0.6)$, while the third brightest images in quad lenses have significantly higher magnifications $(\overline{\mu }=2.6)$. Given the luminosity limit of our quasar sample (M_i < − 20), most quadruply-lensed quasars are brighter than the LSST survey limit. Nonetheless, the image quality of LSST is still critical to the search of these bright quads, since the better depth and spatial resolution are essential for a high completeness and efficiency in the candidate selection.

Figure 8 further illustrates the number of discoverable quaduply lensed quasars as a function of survey depth. We assume an LSST-like PSF size of 07 when counting the number of discoverable lenses, and normalize the numbers to the area of LSST. We also include the number of doubly-lensed quasars and unlensed quasars for comparison. Note that the magnitudes in Figure 8 correspond to the second brightest image for doubly-lensed quasars and the third brightest image for quadruply-lensed ones, in accordance with the definition of discoverable lensed quasars. At m_i ≲ 25, about ∼ 10% − 15% of the discoverable lensed quasars are quads. The magnification bias makes the fraction of quads decreases with survey depth, in agreement with the result in OM10. The number of quad lenses drops quickly at m_i ≳ 26, which is a result of the absolute magnitude cutoff.

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The probability for a deflector to generate a quadruple lens increases with its ellipticity. The right panel of Figure 8 illustrates the distribution of the the deflector ellipticity, e_galaxy, for the CosmoDC2 mock catalog and the one adopted by OM10. OM10 assumes a Gaussian distribution of ellipticity for all the galaxies, while CosmoDC2 use a more complicated approach, where the probabilistic distribution of e_galaxy depends on the bulge-to-disk ratio and the galaxy luminosity. The overall ellipticity distribution of OM10 and CosmoDC2 are similar; correspondingly, the predicted quad fractions at m_i ≲ 26 of this work and OM10 are consistent within statistical errors.

The external shears is another key factor in generating quadruply-lensed systems. Luhtaru et al. (2021) analyzed 39 quadruply-lensed quasars, 15 of which were found to be "shear-dominated." This result illustrates the significant contribution of external shears in making quad lenses. OM10 assumes a log-normal distribution of the total external shear, γ, with a mean of 0.05 and a deviation of 0.2 dex. In comparison, we adopt a redshift-evolving distribution for the external shear. At most redshifts (z ≲ 6), the shear distribution in this work has a lower mean value and a more prominent tail at large shears. We thus expect that the quad lenses in our mock catalog and OM10 have different properties (e.g., lensing configurations). It might be useful to keep in mind that the external convergence and shear distributions used in this work are derived from ray-tracing simulations, which are subject to a number of systematic uncertainties (Section 2.1).

Figure 9 illustrates the distribution of time delays and the magnitude differences between lensed images. The typical time delays are 10¹–10³ days. In most of the doubly imaged lenses, the brighter image arrives earlier than the fainter one, while for quad lenses, the second or third brightest images arrives first in most cases. These results are consistent with the findings in OM10. For a typical time delay of ∼10² days, a cadence of ∼5 days gives a good measurement of the light curves (Suyu et al. 2017). The cadence of LSST survey is not sufficient to provide accurate light curves for most of the lensed quasars. As such, LSST will mainly serve as a deep imaging survey for discoveries, and follow-up monitoring is needed to measure the time delays of lensed quasars.

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Figure 10 shows the distribution of the deflector redshift, z_d , of simulated lensed quasars. The distribution peaks at z_d ∼ 0.7 and drops to a negligible level at z_d ≳ 2.5. This picture is consistent with OM10 and previous modeling of deflector galaxy population (e.g., Hilbert et al. 2008; Wyithe et al. 2011). For comparison, we also include two samples of sources at fixed redshift, z_s = 2 and z_s = 5. Although CosmoDC2 only include galaxies at z < 3, Figure 10 indicates that only a small fraction of lensed quasars have deflector redshift z_d > 3.

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In Figure 11, we present the i-band magnitudes of the deflectors and the deflector-to-quasar magnitude difference of the simulated lensed quasars. In nearly all of the LSST-discoverable lensed quasars, the deflector galaxy is brighter than the LSST limit, which suggests that the impact of the foreground galaxy must be considered when selecting lensed quasar candidates. The deflector-to-quasar magnitude difference spreads over a wide range, and the number of quasar-flux-dominated lenses are comparable to that of galaxy-flux-dominated ones. Correspondingly, we expect a large diversity in observed features for lensed quasars. The galaxy-dominated lenses are likely generated by a massive, luminous deflector and have large image separations, and the quasar-dominated ones usually have compact lensing structures. These objects may require very different candidate selection techniques.

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Section 3.1 shows the impact of QLFs and VDFs on the modeling of the lensed quasar population. In this section, we discuss the systematic uncertainties introduced by our choices of QLFs and VDFs.

At z ≲ 4, the QLFs are well measured down to at least M₁₄₅₀ ∼ − 22 (e.g., Richards et al. 2006; Ross et al. 2013; Palanque-Delabrouille et al. 2016), and these measurements are in good agreement with each other. Since most LSST-discoverable lensed quasars have z_qso < 4, the QLFs adopted by this study provide a good description of the quasar population. Meanwhile, there are still debates about the QLFs at z ≳ 5. A few studies give a faint-end slope of α ∼ − 2 (e.g., Jiang et al. 2016; Yang et al. 2016; McGreer et al. 2018), while others suggest α ∼ − 1.3 (Niida et al. 2020; Kim et al. 2020). In this work, we use the HSC-based high-redshift QLFs that have α ∼ − 1.3. If the faint-end slope turns out to be steeper, there might be more faint lensed quasars at high redshift.

Unlike the QLFs, the deflector VDFs are not well constrained by spectroscopic surveys beyond the local universe. Many previous studies assumes no evolution of the VDF (e.g., OM10; Wyithe & Loeb 2002). However, recent analysis of strong lensing systems indicate that the VDF drops toward high redshift (e.g., Geng et al. 2021), which is consistent with photometric measurements (e.g., Bezanson et al. 2011, 2012) and the CosmoDC2 VDFs used in this work. It is thus critical to use redshift-evolving VDF models when modeling the lens population.

In addition, we consider the uncertainties introduced by the scaling relation used to calculate the velocity dispersion (Equation (2)). Many studies suggest that the M_dyn/M_* ratio evolves with galaxy masses (e.g., Auger et al. 2010; Nigoche-Netro et al. 2016) and redshift (e.g., Beifiori et al. 2014). We thus consider the impact of an evolving M_dyn/M_* ratio in the following form:

$$\begin{eqnarray}&&{\rm{l}}{\rm{o}}{\rm{g}}\displaystyle \frac{{M}_{{\rm{d}}{\rm{y}}{\rm{n}}}}{{M}_{\ast }}=a{\rm{l}}{\rm{o}}{\rm{g}}({M}_{\ast })+b{\rm{l}}{\rm{o}}{\rm{g}}(1+z)+c.\end{eqnarray} \tag{ 5 }$$

This is equivalent to an evolution of $\sigma \propto {M}_{* }^{a/2}{(1+z)}^{b/2}$.

We first set z = 0 to check the mass dependence. We adopt the relation in Auger et al. (2010), which suggests $\mathrm{log}({M}_{* }/{M}_{\odot })=0.885\mathrm{log}({M}_{\mathrm{dyn}}/{M}_{\odot })+0.905$. The left panel of Figure 12 shows the derived VDF at z < 0.1, which does not reproduce the observed local VDF. We thus conclude that adding the mass-dependent term does not gives a good description of the galaxy-velocity dispersion for the CosmoDC2 catalog.

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We then keep the relation M_*/M_dyn = 0.557, and adopt the redshift evolution from Beifiori et al. (2014), which gives σ ∝ (1 + z)^0.12. The right panel of Figure 12 compares the resulted CosmoDC2 VDF and the observed values in Bezanson et al. (2012). At z < 1, the redshift evolution makes little difference compare to the original one, and the resulting VDF is still consistent with observation. At 1.2 < z < 1.5, the redshift-evolving estimation starts to overpredict the number of galaxies at the massive end, where the observed galaxy sample should be complete. This difference might be more significant at z > 1.5 where the redshift evolution term gets larger.

In addition to the redshift evolution of the velocity dispersion, Beifiori et al. (2014) also reports a redshift evolution of the M_dyn/M_* ratio, which follows M_dyn/M_* ∝ (1 + z)^{−0.30±0.12}. By including both trends into Equation (2), we get a very subtle evolution of σ ∝ (1 + z)^−0.03. This result might explain why a non-evolving M_dyn/M_* ratio correctly reproduces the observed VDF. In any case, the redshift dependence is subtle. Given the median redshift of deflectors (z_d ∼ 0.7, Figure 10), the evolution will not strongly influence the resulting lensing statistics.

The CosmoDC2 catalog models dark matter halos down to 10^9.8 M_⊙. According to the M_galaxy–M_halo relation (e.g., Behroozi et al. 2013), a halo of ∼ 10¹⁰ M_⊙ usually has M_galaxy/M_halo ∼ 10⁻³. The CosmoDC2 catalog should thus be complete down to M_* ∼ 10⁷ M_⊙. Our σ-selected deflector sample has a minimum stellar mass of M_* = 10^7.8 M_⊙, which means that the mock catalog is complete for deflectors with σ > 50 km s⁻¹ and z_d < 3.

The main incompleteness comes from the fact that CosmoDC2 catalog only covers 0 < z < 3. Figure 10 suggests that the majority of mock lenses have z_d < 3. Wyithe et al. (2011) studies the strong lensing statistics for z ∼ 8 galaxies, and their results suggest that > 80% lensing systems have z_d < 3. This fraction will be higher for lower source redshifts, and we conclude that our mock catalog is highly complete.

In the discussion above, we adopt the definition of a discoverable lensed quasar in OM10, which requires an image separation ${\rm{\Delta }}\theta \gt \tfrac{2}{3}\times \mathrm{FWHM}$ and a 5σ detection of the second or third brightest lensed image. The motivation of these criteria is that the lensed quasar images need to be at least marginally resolved and barely visible. Under this definition, about one third of the lensed quasars in the mock catalog are not discoverable in LSST. In this subsection, we discuss possible strategies to find these undiscoverable lenses.

Identifying undiscoverable lensed quasars is important because the discoverable criteria can miss some unique and interesting objects. As an example, Fan et al. (2019) report a compact lens with separation of 02 at z = 6.51, J0439+1634, which is the only known multiply-imaged quasar at z > 5 to date. J0439+1634 has a large magnification of μ = 51, making it the brightest quasar at z > 6 in nearly all wavelengths, and provides so far the best case study of a high-redshift quasar. Although J0439+1634 is bright (m_z = 19.49 ± 0.02), the small separation makes it undiscoverable in ground-based imaging surveys. Besides, undiscoverable lenses enable many studies that are impossible otherwise. For instance, small-separation lenses can probe the mass distribution of less massive deflector galaxies.

We first consider the image separation criteria. Figure 6 suggests that LSST-like seeing (∼07) will be sufficient to resolve the majority of lensed quasars. Lensing systems with smaller separation have less massive and thus fainter deflector galaxies. These objects are likely to have quasar-dominated flux with a marginally extended morphology. Since the deflector galaxy and the lensed quasar images are blended, we expect that neither a PSF nor a regular Sérsic profile can provide a good description of their morphology, according to which we can identify these compact lenses. This method is used in Fan et al. (2019) to identify the z = 6.51 compact lensed quasar using ground-based imaging, which is later confirmed by HST imaging.

For lensed quasars that do not meet the second criteria, i.e., the fainter lensed image being brighter than the survey limit, some of them can be discovered by identifying the brighter image as a quasar. If there is a bright galaxy next to it, it is a hint that the quasar might be lensed. Figure 11 suggests that the deflector galaxy is detectable in LSST survey in most lensing systems. Follow-up deep imaging and/or spectroscopy can confirm if there is a fainter lensed image. This technique requires decent image de-blending to disentangle the brighter lensed quasar image and the deflector galaxy.

As we show in Figure 11, the observational features of lensed quasars have a large variety. A complete search of them requires a number of distinct candidate selection methods. The mock catalog can serve as training and testing sets for future surveys of lensed quasars with LSST data.