Issue 
A&A
Volume 590, June 2016



Article Number  A42  
Number of page(s)  14  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201425411  
Published online  04 May 2016 
Multiply imaged quasistellar objects in the Gaia survey
^{1} Extragalactic Astrophysics and Space Observations (AEOS), Institut d’Astrophysique et de Géophysique, Liège University, Allée du 6 Août, 17 (Sart Tilman, Bât. B5c), 4000 Liège, Belgium
^{2} National Astronomical Observatory of Japan (NAOJ), 650 N. A’ohoku Place, Hilo, 96720 HI, USA
email: finet@naoj.org
Received: 26 November 2014
Accepted: 9 October 2015
Aims. We report a study on the statistical properties of the multiply imaged quasistellar objects (QSOs) to be detected within the Gaia survey.
Methods. We considered two types of potential deflectors, the singular isothermal sphere (SIS) and the singular isothermal ellipsoid (SIE), to estimate the number of multiply imaged quasars as well as the normalized distributions of the redshifts of the lensed sources and of their associated deflectors. We also investigated the distribution of the lensing events as a function of their angular size and apparent magnitude. We compared the Gaia survey for multiply imaged quasars to typical groundbased surveys and to an ideal survey that would be carried out with a perfect instrument from space.
Results. Of the 6.64 × 10^{5} QSOs brighter than G = 20 to be detected by Gaia, we expect the discovery of about 2886 multiply imaged sources, 450 of which are expected to be produced by a latetype galaxy. We expect only ~1600 of these multiply imaged quasars to have an angular separation between their images that is large enough to be resolved from seeinglimited observations, and ~80 of them to have more than two lensed images.
Key words: gravitational lensing: strong / quasars: general / cosmological parameters
© ESO, 2016
1. Introduction
The Gaia mission^{1} is currently conducting an allsky coverage for a total duration of five years. The satellite is equipped with three different instruments: an astrometric instrument, a radial velocity spectrometer, and a photometric instrument. The latter produces two lowresolution spectra in the blue and the red, from which the G_{BS}, G_{RS}, and the global Gband magnitudes are derived (see Jordi et al. 2006, 2010 for the description of the photometric system). The survey is expected to be complete down to the Gband magnitude G = 20.
The main goal of the mission is to make a threedimensional map of our Galaxy based on measurements of the photometry, astrometry, and proper motion of ~10^{9} stars. Radial velocity measurements will be obtained for a subsample of brighter objects. The satellite is also expected to detect a very large number of extragalactic objects, some of which will be quasistellar objects (QSOs). The detection of QSOs will be difficult at low Galactic latitudes (b < 25−30°). Judging from the QSO detection in the remaining 60% of the sky, this will probably lead to the detection of 5.5−7 × 10^{5} QSOs (Mignard 2012; Robin et al. 2012; Slezak 2007). This very large sample, in combination with the astrometry precision of the survey of down to ~25 μas, will lead to the direct construction of a new celestial reference frame in the optical that is at least a hundred times denser than the international celestial reference frame (ICRF), which will enable testing general relativity (Mignard 2005). We expect that multiply imaged QSOs are detected among these sources through gravitational lensing by foregrounddeflecting galaxies, which could be detected down to an angular separation of ~0.2′′ (Mignard 2012).
Imaging with Gaia is made by reconstructing 2Dimages from multiple 1Ddrift scan images acquired in different directions (Harrison 2011). As a result of the peculiar driftscan imaging mode of Gaia, it is necessary to properly estimate the properties of the lensed population of QSOs to define a detection strategy for the lensing events. The statistical properties of the QSOs to be detected have been studied by Slezak (2007). In this paper we concentrate on studying the expected statistical properties of the population of lensed QSOs. This information will be used to define the best strategy for detecting these lensing events throughout the different scans generated by the satellite.
Because of the very large number of sources detected by Gaia, we expect the gravitational lenses to constitute an unprecedentedly large sample. In addition to the scientific interest of each individual lensed source, these multiply imaged sources will constitute a statistical sample that may be used to constrain the cosmological mass density parameter Ω_{m} through the statistics of gravitational lensing in the sample, and the lensed sources will also be used to study the evolution in the population of deflecting galaxies.
A previous rough estimation of the number of expected gravitational lenses in the Gaia mission has already been performed (Finet et al. 2012), and Mignard (2008) has studied the effect of gravitational lensing on the reference frame constituted by the sources. We here report an indepth study of their expected statistical properties.
We introduce the mathematical formalism in the next section. We first derive an expression for the probability of a source to be lensed, alternatively modeling the deflectors by spherical (Sect. 2.1.1) and elliptical (Sect. 2.1.2) singular isothermal mass distributions. We then introduce in Sect. 2.2 the joint probability distribution of the Gaia QSOs in the redshiftabsolute magnitude plane that we use in the next subsections to calculate various expected distributions of astrophysical parameters linked to the lensed population. Specifically, the average lensing optical depth in the sample and the redshift distributions of the lensed sources and of the deflectors are derived. Finally, we derive an expression for the distributions of the lensed sources as a function of their apparent magnitude and the angular separation between the lensed images.
In Sect. 3 we present the observational data on which we base our simulations, we derive the bestfit parameter for the QSO luminosity function (LF) evolution models, based on the LF of Richards et al. (2006) and PalanqueDelabrouille et al. (2013), from which we infer the joint probability density of the sources for simulation purposes. Finally, in Sect. 4 we present the results of the simulations and study the effect of the cosmological mass density parameter Ω_{m} and that of the smallest angular separation resolvable in the survey. The latter is done by comparing the results for a perfect instrument, the Gaia observatory, and typical groundbased surveys.
2. Mathematical formalism for gravitationallensing statistics
2.1. Lensing optical depth
Multiple images of a background source arise when a foreground galaxy is located close enough to the source line of sight. In the present work, the sources considered are pointlike QSOs. To calculate the probability for a source to be lensed as a result of a foreground deflector near its line of sight, it is therefore crucial to accurately model the volume density of the potential deflecting galaxies. The comoving volume density of deflectors with a lineofsight velocity dispersion in the range σ to σ + dσ is given by the velocity dispersion function (VDF) , which is modeled by the modified Schechter function (Sheth et al. 2003; Mitchell et al. 2005; Choi et al. 2007; Chae 2010) (1)where Φ_{∗} and σ_{∗} are the characteristic volume density and lineofsight velocity dispersion, α and β are the VDF slope at low and high σ, and is the complete gamma function.
Thanks to their larger mass, earlytype galaxies are more efficient deflectors than latetype galaxies, which tend to form lensed images with a smaller angular separation. The latter, although more numerous, were shown to constitute typically less than 10% of the lensing events in a flux limited sample from groundbased observations (Keeton et al. 1998; Kochanek et al. 2000). Nevertheless, the fraction of lensing events that is due to latetype galaxies increases with a better angular resolution of the survey (e.g., the CLASS survey where ~25% of the lenses are due to spiral galaxies, cf. Browne et al. 2003). Therefore, thanks to the very good angular resolution of the Gaia survey, we expect latetype galaxies to contribute to a significant fraction of the lensed sources. In our simulations, we consider the population of deflectors to be formed by both early and latetype galaxies.
Most of the lensing statistics studies of the evolution with redshift of the deflector population are consistent with a noevolution scenario or very small effect of the evolution (Chae 2010; Oguri et al. 2012). For our estimation we thus neglected the evolution effect in the deflector galaxy VDF and used the value of the VDF parameters measured in the local Universe by Choi et al. (2007), that is, for earlytype galaxies Φ_{∗ ,E} = 8 × 10^{3}h^{3} Mpc^{3}, σ_{∗ ,E} = 161km s^{1} and (α_{E},β_{E}) = (2.32,2.67), and for latetype galaxies Φ_{∗ ,L} = 66 × 10^{3}h^{3} Mpc^{3}, σ_{∗ ,L} = 91.5km s^{1} and (α_{L},β_{L}) = (0.69,2.10).
2.1.1. Singular isothermal sphere deflector
As a first approximation, the total mass distribution of early and latetype galaxies is well modeled by means of the singular isothermal sphere (SIS) profile, that is, by a spherically symmetric mass distribution with a volume density scaling as ∝r^{2}, where r is the distance to the deflector center (see, e.g., Koopmans et al. 2006, 2009 for an observational confirmation of the almost isothermal behavior of the galaxy mass distribution). Such a deflector may lead to the formation of at most two lensed images of a background source, with an angular separation equal to twice the Einstein ring angular radius θ_{E}, which is given by (2)where D_{ds} (respectively D_{os}) is the angular diameter distance between the deflector (respectively the observer) and the source, and c is the speed of light.
A deflector located in the deflector plane (perpendicular to the source line of sight) at a redshift z_{d} will lead to the formation of multiple images of a source at redshift z_{s} with apparent magnitude m if it is located inside an area Σ_{SIS} called the lensing cross section, which is centered on the projected source position and is defined by (Turner et al. 1984) (3)where D_{od} is the angular diameter distance between the observer o and the deflector d. We have introduced the coordinates y = (y_{1},y_{2}), the projection on the deflector plane of the source position, normalized to the scale factor D_{od}θ_{E}, that is, the Einstein radius.
The amplification bias B(m,y) is introduced to take into account the favorable bias in the calculation of the source lensing optical depth, which arises because of the flux amplification in the lensing event, which for instance leads to the inclusion in fluxlimited samples of sources that are intrinsically fainter. In our simulations, we estimated the amplification bias by means of the source differential number counts function (DNCF) as a function of their apparent magnitude m, through the relation(4)where A(y) is the total amplification of the lensing event, that is, the sum of the multiple image amplification moduli.
The integration area S_{y} in Eq. (3) represents the area (normalized to the scale factor) in which the presence of a deflector leads to the formation of a lensing event, defined as the detection of two lensed images by the survey strategy. Depending on the survey angular resolution, the associated 2D integration interval S_{y} varies, and so does the lensing cross section.
The probability for a source with a redshift z_{s} and an apparent magnitude m to be multiply imaged, or the lensing optical depth, is obtained by integrating the density of deflectors (over all possible values of σ) located within the envelope of the lensing cross sections at each intermediate redshift z, defined as the lensing volume (Nemiroff 1989). This leads to the expression (5)where σ is in the range σ_{1} to σ_{2} associated with the deflector population, and cdt/dz is the infinitesimal lightdistance element, which in a flat FLRW universe is expressed as (Peebles 1993) (6)where Ω_{m} is the presentday value of the cosmological mass density parameter. The integration over σ in Eq. (5) may be performed analytically under the assumption of a nonevolving deflector population. Using Eqs. (2) and (3), Σ_{SIS} may be expressed as (7)where . Inserting the latter expression into Eq. (5) and integrating over the σ range [σ_{1},σ_{2}[ → [0, + ∞[, we obtain (8)
2.1.2. Singular isothermal ellipsoid deflector
To reproduce lensing configurations with more than two lensed images, as observed among the known gravitational lens systems, a new model was introduced by Kormann et al. (1994): the singular isothermal ellipsoid (SIE). This model introduces an internal ellipticity into the mass distribution, which is characterized by the axis ratio q between the axes of the projected mass distribution on the deflector plane. The SIE mass profile may produce two, three (cusp configuration), or four lensed images depending on the position of the source relative to the caustics (defined as the lines of infinite amplification in the source plane). Nonsingular mass profiles may produce an additional highly deamplified central lensed image, which is difficult to detect because it is very faint and dimmed by dust extinction in the deflector. Because this central lensed image is not likely to be detected in the Gaia images, we here only consider the lensed images produced by the SIE model that are effectively detected on the CCD frames.
When considering an SIE deflector, the lensing cross section in Eq. (3) now depends on the deflector axis ratio. We may define a lensing event as the detection of multiple images or as the detection of a given number i of lensed images (two, three, or four), regardless of whether we account for the ability of the instrument to resolve lensed images with a too small angular separation. Consequently, depending on the definition of the lensing event, the area S_{y} in Eq. (3) varies, and we therefore define different cross sections: Σ_{SIE} corresponds to the detection of multiple images irrespective of their number, and Σ_{SIE,i} to the detection of i lensed images.
The volume density of the deflectors is now also a function of the axis ratio q. Because of the lack of observational constraints on the q−σ correlation for the deflectors, we assume that the distribution of the axis ratio is independent of the deflector lineofsight velocity dispersion. The number density of deflectors with a lineofsight velocity dispersion and axis ratio in the range σ to σ + dσ and q to q + dq, respectively, is thus (9)where n_{q}(q) is the normalized distribution as a function of the axis ratio q for early or latetype galaxies, and where the deflector VDF is given by Eq. (1).
Koopmans et al. (2006) and Sluse et al. (2012) have independently confirmed through studying various gravitational lens samples that elliptical galaxy isophotes and the isodensity curves of their projected mass distribution have wellcorrelated ellipticities and majoraxis directions. The normalized distribution n_{q}(q) can thus be estimated from the distribution of the isophotes of earlytype galaxies as measured by Choi et al. (2007) in the local Universe.
To calculate the lensing optical depth for a source with deflectors modeled by SIE mass profiles, we need to integrate over both variables σ and q (Huterer et al. 2005), which leads to (10)The integration over σ can be performed using Eq. (7) while adopting the same assumptions as for the SIS case. This leads to (11)We have developed software toolboxes using Matlab, allowing us to calculate the lensing cross sections and optical depths by modeling the deflectors by means of both the SIS and the SIE mass distribution (Eqs. (3), (5), and (11)).
2.2. QSO joint probability density d
The sources detected in a survey are characterized by their absolute magnitude M and their redshift z_{s}. Each source has a probability to be detected with a redshift and an absolute magnitude in the ranges z_{s} to z_{s} + dz_{s} and M to M + dM, respectively. Furthermore, we may define the joint probability density associated with by means of the relation (12)For an already existing survey, may be estimated from the normalized smoothed histogram in the plane of the detected sources. However, for prospective simulation purposes, can be estimated by means of the QSO luminosity function (Oguri & Marshall 2010) by means of the relation (13)where

accounts for selection biases in the plane, which occur during the detection procedure of the sources. For a perfect fluxlimited sample, this function equals one in the region ofthe plane which leads to an apparent magnitude brighterthan the survey limiting magnitude.

N _{QSO} is the total number of QSOs detected within the survey;

d V _{c} /d z is the differential contribution at redshift z to the total comoving volume accessible by the survey, which in a flat expanding FLRW universe may be expressed as (14) where D _{c} is the lineofsight comoving distance at redshift z , its differential contribution, and Ω_{Gaia} is the solid angle covered by the survey in which QSO detection is possible.
The joint probability density d_{obs}(z_{z},M) is closely related to observable distributions of the source population. Indeed, the marginal distribution obtained by integrating over M is the normalized redshift distribution of the sources. Similarly, the differential number counts function (DNCF) as a function of the apparent magnitude may be estimated by (15)where and are the distance modulus and Kcorrection at redshift z_{s}, respectively.
2.3. Source statistical properties
The joint probability density of the source population is directly linked to the distribution of the sources in the plane and to their observable distributions n_{z} and n_{m}. It may be used as a weighting factor to estimate the mathematical expectation of any function of z_{s} and M.
We assume that we have an expression for the lensing optical depth as a function of z_{s} and M for a given source. This expression is trivially obtained from , and (Eqs. (5) and (11)) considering . Its mathematical expectation ⟨τ⟩ is simply given by (16)where ⟨τ⟩ represents the fraction of sources in the detected population to have undergone a gravitational lensing event.
Similarly, the expected normalized redshift distribution of the deflectors is given by (Oguri et al. 2012) (17)and the expected normalized redshift distribution of the lensed sources by (Oguri & Marshall 2010) (18)By integrating over the source redshift, we may also derive the normalized distribution as a function of the apparent magnitude m of the lensed sources (19)Finally, we define the normalized probability density ω_{θE} of observing a gravitational lens system with an angular configuration θ_{E} in the lensed population. From the definition of θ_{E} in Eq. (2), we may trivially derive the following relations (20)where . Furthermore, from the definition of the lensing cross section in Eq. (3), we have (21)where is the lensing cross section evaluated for the typical value of θ_{E} = θ_{∗}. Inserting Eqs. (20) and (21) in the definition of τ_{SIS} in Eq. (5) and making use of Eq. (1), we may derive the expression of τ_{SIS} for which the integration is done over θ_{E} rather than σ. Deriving this expression with respect to θ_{E}, we find (22)The normalized probability density ω_{θE} is then simply obtained by averaging the previous expression over the detected population of sources (23)Using the joint probability density as a weighting factor to average quantities over the entire population of sources detected in the survey, we have derived expressions for the mean optical depth ⟨τ⟩ (Eq. (16)), and of the normalized redshift distributions expected for the deflectors ω_{zd} (Eq. (17)) and for the lensed sources ω_{zs} (Eq. (18)). We also derived the normalized distributions ω as a function of the apparent magnitude m and Einstein angle θ_{E} of the lensed sources (Eqs. (19) and (23), respectively).
The fundamental quantity needed is the joint probability density corresponding to the expected Gaia survey sources, which we estimate in the next section.
3. Observational constraints
In this section, we describe the observational constraints used to estimate the joint probability density for the Gaia survey.
3.1. QSO luminosity function
Fig. 1 Luminosity function as a function of the absolute magnitude M_{i} in the SDSS iband for different redshift bins. We have included the LF determined by Richards et al. (2005) and PalanqueDelabrouille et al. (2013) (converted to M_{i} magnitude). The different fitted models correspond to no contraints on the redshift behavior of M_{∗} and Φ_{∗} (dashed light gray line), to an evolution model for Φ_{∗} and M_{∗} is fitted freely (light gray line), and to M_{∗} and Φ_{∗} both constrained by an evolution model (continuous black line). The latter is the final evolution model chosen for the simulations, see main text for the full description. 

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Thanks to Eq. (13), may be estimated through the LF of the observed sources. We need an estimate of the QSO LF and its behavior with redshift, spanning over the entire redshift and absolute magnitude ranges probed by the Gaia Survey. Various evolution models of the QSO optical LF have been proposed (e.g., Richards et al. 2005, 2006; PalanqueDelabrouille et al. 2013; Ross et al. 2013 for some recent works). Unfortunately, none of them fully spans the entire redshift and absolute magnitude ranges accessed by the Gaia Survey.
In the remainder of this section, we consider the binned luminosity function derived by Richards et al. (2006) as a function of the M_{i} magnitude (based on the SDSSDR3) and that derived by PalanqueDelabrouille et al. (2013) in the SDSS gband based on the SDSSIII and the MMT data, and we fit an evolution model on the combined data. PalanqueDelabrouille et al. (2013) have derived the binned LF versus the M_{g} continuum absolute magnitude associated with the SDSS gband, with the zero point of the continuum Kcorrection at z = 2. For the conversion from M_{g} to M_{i}, we follow Ross et al. (2013) and convert through(24)This transformation is derived assuming the continuum to be a single powerlaw with a spectral index α_{ν} = −0.5, which they used to define the continuum.
The derived binned luminosity functions from Richards et al. (2006) and PalanqueDelabrouille et al. (2013) are depicted in Fig. 1, both expressed as a function of the M_{i} magnitudes for the different redshift bins. Shen & Kelly (2012) have also derived the QSO LF from the SDSSDR7 in the same redshift and apparent ranges as Richards et al. (2006) and their results agree very well. We used the result from Richards et al. (2006) because the redshift bins are similar to those of PalanqueDelabrouille et al. (2013).
We followed Richards et al. (2006) and used absolute magnitudes M of the continuum, with a zero point of the continuum Kcorrection at redshift z = 2. We determined the Kcorrection for the Gaia Gband using the Gband spectral transmission function (Jordi et al. 2010) with the QSO synthetic spectrum derived by Vanden Berk et al. (2001) from SDSS QSO spectra. We define the continuum absolute magnitude and derive the Gaia Gband Kcorrection in Appendix A.
To facilitate the comparison with previous results, we determined the LF evolution considering absolute magnitudes M_{i} in the SDSS iband. We converted the absolute magnitudes M_{i} to the Gaia Gband absolute magnitude M using the transformation law derived in Appendix A, based on the QSO spectrum from Vanden Berk et al. (2001), the Gband Kcorrection previously described, and SDSS iband Kcorrection given in Richards et al. (2006).
To model the LF, we used the conventional double powerlaw form for the QSO LF in terms of absolute magnitudes (25)where M_{∗} and Φ_{∗} are the characteristic absolute magnitude and number density, respectively, and α and β are the bright and faint end slopes of the LF, respectively.
For the slope parameters, we considered β = −1.45 and α = −3.31 (values taken from Richards et al. 2005 from the analysis of the combined SDSS and 2dF samples) over the entire redshift range.
We first fit the combined data in each redshift bin assuming constant slope parameters α and β, fitting Φ_{∗} and M_{∗} without any evolution model constraints on their behavior as a function of the redshift. The resulting LF is shown as a dashed light gray curve in Fig. 1, and the values of the bestfit parameters are shown in Fig. 2 as light gray circles for the different redshift bins. The rather large error bars arise because Φ_{∗} and M_{∗} are highly correlated.
Fig. 2 Behavior of the LF Φ_{∗} and M_{∗} parameters as a function of the QSO redshift. The dark continuous line shows the fit model used in our simulations. The light gray markers show the best fit parameters for Φ_{∗} and M_{∗} when fitting the LF separately in each redshift bin, without any evolution model constraints. When constraining the evolution of the characteristic density, the best parameters are found to be log Φ_{∗ ,lowz} = −5.85, α_{Φ∗} = −0.77 ± 0.31, and z_{ref} = 2.09 ± 0.28. 

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Following the results of Ross et al. (2013), we assumed that log Φ_{∗} is constant for z< 2.05 (z< 2.2 in Ross et al. 2013) and that it can be fitted as linearly evolving for higher z. We thus assumed an evolution model for Φ_{∗} given by (26)In each redshift bin we then fit the value of M_{∗} for each Φ_{∗} given by the model. The bestfit parameters are log Φ_{∗ ,lowz} = −5.85, α_{Φ∗} = −0.77 ± 0.31, and z_{ref} = 2.09 ± 0.28.
The behavior of Φ_{∗} as a function of the redshift is shown in the lower panel of Fig. 2, and the corresponding fitted values of M_{∗} are shown as dark gray markers on the upper panel.
Finally, motivated by the smooth redshift evolution of M_{∗}, we fit the evolution of M_{∗} by a thirdorder polynomial. The fit is shown in the upper panel of Fig. 2. The bestfit parameters are [c_{3},c_{2},c_{1},c_{0}] = [−0.0427,0.5484,−2.7563,−22.8766].
The final evolution model used for M_{∗} and log Φ_{∗} are shown as a function of the redshift, as a continuous dark gray line in Fig. 2. The corresponding LF is represented for the different redshift bins in Fig. 1 as a continuous dark gray line. We represent for each redshift bin the faintest sources detectable by Gaia assuming G = 20 is the faintest magnitude achievable. The limit is shown as a vertical light gray dashdotted line. The conversion from G to M_{i} is made using the color evolution as a function of the redshift, described in Appendix A, while adopting the SDSS iband Kcorrection.
In the range accessible to the Gaia mission, our LF evolution model fits the data over the entire redshift range very well.
3.2. Joint probability density
To estimate the joint probability density for the Gaia survey, we used the QSOLF function derived in Sect. 3.1 to compute through Eq. (13).
Fig. 3 Left: joint probability density for the Gaia sources, derived using the LF evolution model described in the text. Right: DNCF as a function of the Gband magnitude. We use a combined observational sample of the SDSSDR3 and 2QZ/6QZ for magnitudes brighter (fainter) than i ~ 19, converted to Gband magnitudes and thus assuming the DNCF shape in Gband to be similar. We also show the fit used to estimate the DNCF. 

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Fig. 4 Left: average lensing optical depth as a function of the survey angular resolution θ_{min} for early and latetype galaxy deflectors. We modeled the deflectors as SIS (⟨τ_{SIS}⟩continuous black line) and SIE mass distributions (⟨τ_{SIE}⟩continuous gray line). In the latter case, we also show the average optical depth with detection of two lensed images (⟨τ_{SIE,2}⟩ – dashed gray line). Right: fraction of lensed sources detected with three (light gray) or four (dark gray) lensed images as a function of θ_{min}. The results for the case of early and latetype galaxy deflectors are shown with continuous and dashdotted lines, respectively. In both panels we indicate the typical value of θ_{min} for the Gaia survey and typical seeinglimited groundbased observations. 

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We limited the redshift range to 0 <z ≲ 4.5. Although the spectrophotometric imaging of Gaia will enable detecting QSOs with a higher redshift, the limiting magnitude of G = 20 corresponds to very bright QSOs at redshift higher than z ~ 4.5, which are very rare.
The resulting is shown in the plane in the left panel of Fig. 3. The gray scale is proportional to the probability of detection, a darker gray indicating a higher probability of detection. For clarity, we indicate the magnitude cutoff of the survey G = 20 as well as the brighter cutoff G = 16 that was imposed because of the scarcity of such bright QSOs. For sources fainter than G = 20, d_{obs} is null because the detection probability of these sources is assumed to be null.
To assess the ability of the derived to reproduce the observed properties of the real sources, we compared the DNCF as a function of the apparent magnitude derived with Eq. (15) to observational sets that were used to determine the LF by applying the magnitude and redshift cuts of the observational samples to the joint probability density.
Richards et al. (2006) have estimated the QSO DNCF in the SDSS iband on the basis of the SDSSDR3 QSO catalog for sources restricted to the redshift ranges 0.3 <z< 2.2 and 3 <z< 5. In the lower redshift range, the survey is complete down to i ≃ 19. For the fainter magnitudes, we used the DNCF from the 2SQ/6QZ survey (also given in Richards et al. 2006). The different data sets are displayed in the right panel of Fig. 3.
We derived the DNCF from through Eq. (13) by restricting the redshift range to 0.3 <z< 2.2 and 3 <z< 5. The results are shown in the right panel of Fig. 3 as continuous and dashed light gray curves. The simulated DNCF agree very well with the observational data. Following Richards et al. (2006), we used as definition of a QSO a source with an absolute magnitude in the continuum of the SDSS iband (with zero point at z = 0) brighter than −22.5, considering a flat expanding universe with H_{0} = 70 km s^{1} Mpc^{1} and Ω_{m} = 0.3. In the same figure, we represent with a dark gray line the expected DNCF of the entire Gaia population that we used in Eq. (4) to calculate the amplification bias.
Comparison of the mean lensing optical depth and the expected number of detected multiply imaged quasars for different values of the survey angular resolution corresponding to a perfect survey (θ_{min} = 0′′), Gaia (θ_{min} = 0.2′′), and typical groundbased observations (θ_{min} = 0.6′′).
Combining Eqs. (13) and (14), the normalization factor of the joint probability density is Ω_{Gaia}/N_{QSO}, that is, the number of QSOs detected per steradian, from which we may trivially derive the total number of QSOs expected to be detected in the survey. Considering that QSOs will be detectable over 60% of the sky (thus excluding the low galactic latitude fields), this leads to the expected detection of 6.64 × 10^{5} sources brighter than G = 20. Mignard (2012) has estimated the surface density of QSOs brighter than G = 20 and concluded that the Gaia survey should expect the discovery of 5.5 to 7 × 10^{5} QSOs; this result agrees well with our own estimate. Slezak (2007) also previously estimated the number of QSOs to be detected: 7.2 × 10^{5}.
4. Results
We have computed the mean lensing optical depth ⟨τ⟩ for the whole population of QSOs by means of Eq. (16) for different values of the survey angular resolution θ_{min} (i.e., the minimum image separation for which pointlike sources with similar amplification may be distinguished). In Eq. (16), we alternatively considered the lensing optical depth τ for deflectors modeled as SIS (τ_{SIS} in Eq. (5)) and as SIE deflectors (Eq. (11)), considering the population of both early and latetype galaxies. When considering SIE deflectors, we calculated both τ_{SIE} and τ_{SIE,i}. In Fig. 4 we represent as a function of the angular resolution θ_{min} of the survey the dependence of the different average optical depths ⟨τ_{SIS}⟩, ⟨τ_{SIE}⟩ and ⟨τ_{SIE,i}⟩ and the evolution of the fraction of events with formation of three or four lensed images, that is, ⟨τ_{SIE,3}⟩/⟨τ_{SIE}⟩ and ⟨τ_{SIE,4}⟩/⟨τ_{SIE}⟩, for both types of galaxy populations.
The average total optical depths ⟨τ_{SIS}⟩ and ⟨τ_{SIE}⟩ show a very similar behavior: they both decrease as θ_{min} increases because lensing events with images too close to each other are not resolved by the survey and are detected as single sources. In Table 1 we list the numerical values of the lensing optical depth for different values of the θ_{min} parameter, representative of a perfect survey (θ_{min} = 0′′), the Gaia survey (θ_{min} = 0.2′′), and seeinglimited groundbased observations (θ_{min} = 0.6′′). θ_{min} represents the smallest angular separation between two pointlike sources with similar brightness for which the survey source detection procedure is capable to separate the two lensed images. We considered θ_{min} to be independent of the relative brightness between the lensed images. This assumption is motivated by the fact that the regions contributing the most to the lensing cross sections are the most amplified (where the source is located inside and near the tangential caustics). For these configurations, the brighter and closer lensed images are located near the tangential critical curve and show similar amplification (see, e.g., Schneider et al. 1992, Chap. 6). For typical groundbased observations, we considered seeinglimited observations (typically ~1−1.2′′). With PSFfitting techniques, we may at best resolve pointlike images with the same brightness separated by half the PSF full width at half maximum, that is, by half of the seeing value.
Fig. 5 Left: expected normalized distributions ω as a function of the redshift of the lensed sources (z_{s}) and of the deflectors (z_{d}). For comparison, we also include the normalized redshift distribution of the sources (ω_{QSO}). The simulations were made assuming θ_{min} = 0.2′′. We indicate the median value of each distribution (, and ). Right: normalized distribution of the lensed sources as a function of the apparent magnitude. For comparison, we show the DNCF as a function of the magnitude of all the detected sources, normalized by N_{QSO}/ Ω_{Gaia}. 

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For the earlytype galaxy population, the average lensing optical depth for a perfect survey is 3.994 × 10^{3} (3.747 × 10^{3} for the SIE lens model), and this is only a few percent lower for the Gaia survey, that is, 3.917 × 10^{3} (3.663 × 10^{3}). For seeinglimited groundbased surveys, however, about onethird of the lensed sources are unresolved, with the mean lensing optical depth dropping to 2.718 × 10^{3}(2.431 × 10^{3}). Considering the estimated number of 6.64 × 10^{5} sources to be detected by the Gaia survey, there are thus 2653 (2490) expected lensed sources, out of which 2602 (2433) should be detected by the Gaia survey. On the other hand, the seeinglimited groundbased followup of the lensing events will only be possible for 1806 (1615) of these sources, unless adaptive optics observations are made possible using large telescopes.
Although the latetype galaxy populations are more numerous, they are less efficient deflectors because of their lower mass. For the Gaia survey, the average lensing optical depth due to the latetype galaxy population is 0.84 × 10^{3} (0.68 × 10^{3} for the SIE lens model), but because they tend to form lensing events with lensed images closer to one another, they represent a negligeable fraction for typical seeinglimited groundbased observations. This deflector population is expected to lead to the formation of 558 (453 for the SIE lens model) multiply imaged quasars.
In conclusion, combining the expectation for the early and latetype galaxy populations, the Gaia survey is expected to lead to the detection of 3160 (2886 for the SIE lens model) multiply imaged quasars.
These results are consistent with a former simplified estimate of the number of gravitational lens systems to be detected by Gaia by Finet et al. (2012), who found ⟨τ⟩ = 5.9 × 10^{3} for the SIS deflectors without taking into account the finite instrument resolution and considering only the population of earlytype galaxies. The authors used a flat FLRW universe with Ω_{m} = 0.27 and H_{0} = 72 km s^{1} Mpc^{1}. Considering the same universe model and a perfect instrument, we find ⟨τ_{SIS}⟩ = 4.3 × 10^{3}. The slight differences probably arise because they considered sources as bright as G = 15 to be affected by a larger amplification bias and a redshift range up to z = 5, which slightly overestimates the very high redshift sources that have a higher optical depth.
⟨τ_{SIS}⟩and ⟨τ_{SIE}⟩ lead to a very similar estimate of the number of lensed sources to be detected and show a similar behavior as a function of the angular resolution parameter θ_{min}, as seen in Fig. 4. Nevertheless, the SIS model leads to a lensing optical depth estimate about 10% higher than that of the SIE model. This is because the SIE mass distribution, initially introduced to conserve the projected mass inside isodensity curves with respect to the SIS case (Kormann et al. 1994), does not preserve the size of the geometrical cross section, that is, the area inside the caustics. For very elliptical deflectors, the geometrical cross section of the SIE deflector is smaller than that of the SIS because the radial caustic curve flattens when the ellipticity increases. Furthermore, the probability for a lensed source to have a given total amplification is different for the two mass distributions, thus leading to a different amplification bias effect (Huterer et al. 2005).
Figure 4 shows that of the gravitationally lensed sources, the configurations with two detected images are the most likely. Furthermore, the fraction of events with two lensed images increases as θ_{min} increases, from 91.7% for a perfect survey to 94.8% for θ_{min} = 0.6′′ (cfr Table 1) for the earlytype galaxy deflectors, because some configurations with three or four formed images have only two resolved images. We see a similar behavior for the results considering the latetype deflectors. We represent in Fig. 4 the evolution of the fraction of events with three or four detected lensed images as a function of θ_{min}. From Table 1, considering the earlytype galaxy deflectors, we see that 204 lensed sources are expected to be composed of four lensed images (three with three lensed images), but because of the limited angular resolution of the satellite, only 134 (62) are expected to be detected with four (three) observable lensed images. Of these, 75 (9) lensing events with three (four) lensed images will be observable by means of seeinglimited groundbased observations, which will complement the times series and spectra acquired by the satellite. The latetype population is expected to lead to the formation of 57 events with more than two lensed images among the lensed sources in the Gaia survey, none of which could be detected in a seeinglimited groundbased survey.
The lensed sources to be detected by the Gaia mission will thus constitute an unprecedented statistical sample, at least an order of magnitude larger than the existing ones, such as the groundbased SDSS Quasar Lens Search sample (62 lensed sources, 26 in the statistical sample, see Inada et al. 2012) or the 13 lenses from the CLASS statistical sample (Browne et al. 2003).
In the left panel of Fig. 5, we show the expected normalized redshift distributions of the lensed sources and of the deflectors . We computed these distributions using Eqs. (17) and (18), assuming the typical angular resolution of Gaia θ_{min} = 0.2′′. For the normalized deflector redshift distribution, we represent the expected distributions for the early and the latetype deflector populations. For comparison, we represent the normalized redshift distribution of the sources (ω_{QSO}) obtained by integrating over M. As the latter is a normalized joint distribution, its marginal distribution ω_{QSO} is normalized as well.
Fig. 6 Left: normalized redshift distribution of the deflectors and its cumulative representation for three different values of the angular resolution θ_{min} = 0,0.2 and 0.6′′. Right: cumulative distribution τ^{1}dτ/dz as a function of the lens redshift. We show the case for two different source redshifts (z_{s} = 2 and 4) and three different values of the angular resolution parameter (θ_{min} = 0,0.6 and 1′′), modeling the deflectors by means of the SIS and SIE models. 

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We also indicate the median value of each distribution (, , and ). We clearly observe a shift for the lensed population toward higher redshifts compared to the entire population of QSOs, with a distribution shifting from to . This is mainly explained by the increase in geometrical lensing volume with the source redshift, which introduces a favorable bias toward sources located farther away. The shape of ω_{zs} is also influenced by the amplification bias, which in our case favors lowredshift sources. Because these sources have on average a brighter apparent magnitude (than those with a higher redshift), their amplification bias calculated through Eq. (4) is larger because of the steeper slope of the DNCF for these magnitudes.
The normalized redshift distributions of the deflectors are different for the early and the latetype galaxy deflectors. For a perfect instrument (i.e., θ_{min} = 0′′), these distributions would be identical, as demonstrated in Appendix B. But in this simulation we considered θ_{min} = 0.2′′. This excludes lensing events with lensed images whose image separation is too small. Therefore, the fraction of rejected lensing events is higher for the latetype galaxy population. Furthermore, for a given source redshift, the first lensing events to be rejected are those with a higher deflector redshift (because the Einstein angular radius scales as ). This leads to a normalized deflector redshift distribution that peaks at lower redshift for the latetype galaxy population.
The normalized distributions as a function of redshift of the lensed sources and of the deflectors show no dependence on the chosen deflector model: both the SIS and SIE models lead to exactly the same distributions. Furthermore, the normalized redshift distribution of the lensed sources is identical when considering early or latetype galaxy populations.
In the right panel of Fig. 5 we illustrate the normalized distribution of the lensed sources as a function of their apparent magnitude computed from Eq. (19) for the SIS lens model. For comparison, we show the DNCF as a function of magnitude of all the detected sources, normalized by N_{QSO}/ Ω_{Gaia}, the number of sources detected per solid angle in the magnitude range 16 <m< 20. The distribution of the lensed sources presents an excess of brighter sources, benefiting from the amplification bias. In our simulations, we found no effect of the angular resolution θ_{min} on . Neither did we find differences between the ω_{m} distributions when considering the early and latetype deflector populations.
We now analyze the effect of the finite instrumental resolution on these different distributions. We showed that modeling the deflectors by means of the SIE and SIS mass distributions leads to exactly the same normalized distributions and , therefore we only considered the SIS deflectors in the remaining part to minimize the computation time. In the left panel of Fig. 6 we display the normalized distribution of the deflectors and its cumulative function as a function of the deflector redshift for three different values of the angular resolution θ_{min} corresponding to the perfect instrument case, the Gaia mission, and typical groundbased observations.
The normalized and cumulative distributions corresponding to θ_{min} = 0′′ and θ_{min} = 0.2′′ look similar. When compared to the groundbased observations (θ_{min} = 0.6′′), we conclude that the effect of the loss in resolution power is to miss the lensing events with a deflector at higher redshift. This may be easily understood for the SIS case. An SIS deflector produces two lensed images separated by an angle equal to twice the Einstein angle θ_{E}. From the definition of the Einstein angle in Eq. (2), θ_{E} scales as (27)For a source at a redshift z_{s}, is a decreasing function of the deflector redshift, consequently, θ_{E} decreases as the deflector redshift z_{d} increases, which will produce lensing events with images closer to each other, the first to be discarded as θ_{min} increases.
Fig. 7 Left: average optical depth ⟨τ⟩ (considering all deflector types) as a function of cosmological matter density parameter Ω_{m}, modeling the deflectors with the SIS and SIE models. Right: normalized redshift distributions ω_{zs} and ω_{zd} of the lensed sources and of the deflectors (for the earlytype galaxy population) for different values of Ω_{m} = 0,0.3 and 1. All simulations were produced for θ_{min} = 0.2′′. 

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We now consider the cumulative distribution as a function of the deflector redshifts. For simplicity, we only considered here the earlytype galaxy population. For the groundbased observations, 90% of the observed lensed sources have a deflector with a redshift lower than z = 0.88. Most of the studies of the evolution effect in the deflector population based on lensing statistics are compatible with a noevolution scenario or very little evolution of the deflectors (Oguri & Marshall 2010; Chae 2010; and Oguri et al. 2012). But it is very inefficient to use groundbased observations (from which most of the statistical samples of lenses are issued) to study the evolution of the deflector population because they cut out all lens systems with a deflector at high redshift and are thus only suitable for studying the lowredshift population (typically, z< 0.88).
For the Gaia mission, ~30% of the detected lenses that are due to the earlytype galaxy population will have a deflector at a redshift higher than z> 0.88, that is, ~800 lenses assuming the 2433 lenses to be detected, and ~240 lensed sources will have a deflector in the redshift range 1.27 <z_{d}< 2. The statistical sample of lenses to be unveiled by the Gaia mission will thus provide a sample that is very well suited for the evolution study of the population of early and latetype galaxies at high redshift because of both the very large number of sources and the high angular resolution power with which the lensed sources are detected.
We also computed the normalized distribution of the lensed sources as a function of redshift for different values of θ_{min}. The angular resolution parameter θ_{min} has no effect at all on . This implies that the relative decrease in the lensing optical depth due to an increase in θ_{min} is independent of the source redshift and absolute magnitude. To understand this, we computed the cumulative distribution of dτ/dz_{d} normalized to the source optical depth for a perfect survey, that is, (28)The behavior of the cumulative distribution in Eq. (28) as a function of redshift is shown in Fig. 6. We show two different sources at redshift z_{s} = 2 and 4, considering three different angular resolutions θ_{min} = 0,0.6 and 1′′, and modeling the deflector with both the SIS and SIE deflectors. For clarity, we do not show the case for θ_{min} = 0.2′′.
We first consider θ_{min} = 0′′ (shown as a black dashed line). For this case, the normalized cumulative distributions are identical for the SIS and SIE cases. Because the distributions corresponding to the different z_{s} differ, the differential contribution to the lensing optical depth is different for the two sources. Nevertheless, for the θ_{min} = 0.6′′ instrumental resolution we observe that the fraction of the optical depth lost due to an increase in θ_{min} is the same for z_{s} = 2 and 4 for both the SIS and SIE cases. The fraction of the optical depth lost due to an increase in θ_{min} is thus independent of redshift, or in other words, the ratio (29)is independent of the source redshift. We computed the same distribution for sources with different apparent magnitudes and found no effect of the source magnitude on the distribution in Eq. (28).
As a consequence, the behavior as a function of θ_{min} of the average optical depth ⟨τ⟩ can be computed by considering a perfect instrument and calculating the ratio in Eq. (29) for a source with a randomly chosen redshift and magnitude.
4.1. Effect of Ω_{m}
In the left panel of Fig. 7 we show the evolution of the average lensing optical depth ⟨τ⟩ as a function of the cosmological matter density parameter Ω_{m}, assuming a flat FLRW universe and considering the finite angular resolution θ_{min} = 0.2′′ corresponding to the Gaia survey. We observe a very high dependence of the average optical depth as a function of Ω_{m} with an order of magnitude difference between an empty universe and a universe full of matter (Ω_{m} = 0 and 1, respectively). For this reason, using the fraction of lensed sources in a sample of sources has been proposed by Turner et al. (1984) to constrain the value of Ω_{m}.
The observed behavior agrees very well when modeling the deflectors by means of the SIS and SIE deflectors, although the SIS model always slightly overestimates the lensing optical depth.
In our simulations for the SIE deflector, we observed no dependence as a function of Ω_{m} of the relative fraction of lensing events on a given number of lensed images.
In the right panel of Fig. 7 we show the expected normalized redshift distribution of the lensed sources and of the deflectors for the three different values of Ω_{m} = 0,0.3 and 1. For the sake of clarity, we only show the deflector redshift distribution corresponding to the earlytype galaxy population, but the one associated with the latetype population presents a similar behavior. As previously mentioned, the obtained normalized redshift distributions ω_{zs} and ω_{zd} are identical for the SIS and SIE models. To minimize the computing time, we present the distributions obtained for the SIS model case.
Finally, in Fig. 8 we represent the normalized distribution ω_{θE} as a function of Einstein angle θ_{E} of the lensed sources for different values of Ω_{m}. These distributions were obtained with the help of Eq. (23). We considered the deflectors to be modeled with SIS deflectors and an angular resolution θ_{min} = 0.2′′ corresponding to the Gaia survey. We find a very small dependence of on the cosmological model parameter value.
We also computed the normalized distribution as a function of apparent magnitude m and found no effect of Ω_{m}.
Fig. 8 Effect of the cosmological matter density parameter Ω_{m} on the normalized Einstein angular radius distribution of the lensed sources. We considered the deflectors to be modeled by SIS deflectors and an angular resolution θ_{min} = 0.2′′ corresponding to the Gaia survey. θ_{E} is expressed in arcsecond. 

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5. Conclusions
Of the 6.64 × 10^{5} QSOs that are brighter than G = 20 to be detected in the Gaia survey, we expect about 2886 to be multiply imaged sources and that 450 of these are produced by a latetype galaxy. We have modeled the deflector population by means of the SIE and SIS mass distributions and found both model predictions to agree very well, although the SIS model overestimates the mean lensing optical depth by ~10%. Most of the multiply imaged sources will be composed of two images, but we expect more than 250 lensed sources with more than two lensed images detected.
We expect only ~1600 of the multiply imaged quasars to have an angular separation between their images that is large enough to be resolved from seeinglimited observations (i.e., considering a groundbased survey without an adaptive optics system), allowing the acquisition of groundbased data to complement the spectra and time series provided by the satellite, ~80 of them having more than two lensed images detected.
We showed that lenses with a deflector at high redshift tend to be missed as the angular resolution of a survey becomes worse because these events are characterized by a smaller Einstein angular radius. Thanks to its angular resolution of θ_{min} = 0.2′′, the lensed sources discovered in the Gaia survey will thus provide a unique statistical sample of lensed sources from which to study the evolution effects of the deflecting galaxy population, with the detection of ~800 lenses at a redshift between 0.8 and 2.
We did not consider the influence of the deflector environment, which may produce an additional shear and convergence to the gravitational potential. Oguri et al. (2005) have shown that the additional convergence produced by the galaxy environment may increase the lensing probability (especially at large angular separation) by boosting the image separation and amplification bias, mainly driven by convergence. Huterer et al. (2005) showed that the external shear increases the fraction of quads in a sample of lensed sources. Our estimate of the number of lenses to be discovered by Gaia may thus slightly underestimate the lenses and quads to be discovered. We also neglected multiple deflectors at different redhshifts because the probability for such an event would be on the order of ⟨τ⟩^{2} ~ 10^{5}, which is negligible compared to the lensing probability by a single deflector.
We computed the normalized redshift distributions of the lensed sources ω_{zs} and of the deflectors ω_{zd} and found that these normalized distributions are the same whether the deflectors are modeled by means of the SIS or the SIE mass distributions. The normalized deflector redshift distribution expected for the latetype galaxy population peaks at a higher redshift than that for the earlytype one.
Furthermore, ω_{zs} is independent of the angular resolution of the survey. As a consequence, we conclude that the fraction of the optical depth lost by a source when increasing θ_{min} is independent of source redshift.
Finally, we analyzed the effect of the cosmological matter density parameter Ω_{m} on the average lensing optical depth and on the distributions as a function of the redshift of the lensed sources and deflectors. We conclude that all three are sensitive to the cosmological model parameter value and may be used to constrain the cosmological model. We found no effect of Ω_{m} on the fraction of lensed sources as a function of the number of lensed images or on the apparent magnitude distribution of the lensed sources.
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Appendix A: Gaia Gband Kcorrection and magnitude conversion
We here describe the different relations used to convert absolute magnitudes between the Gaia and the SDSS photometric systems.
We followed Richards et al. (2006) and express the absolute magnitudes M in the Gaia Gband as magnitudes of the continuum, with a zero point of the continuum Kcorrection at redshift z = 2. M is thus defined by (A.1)where K_{em} is the contribution from the emission lines and K_{cont,z = 2} the continuum Kcorrection with zero point at redshift z = 2.
Fig. A.1 Kcorrection in the Gaia photometric Gband as a function of redshift. We represent the total Kcorrection (K) with zero point of the continuum contribution at z = 2, the continuum contributions with zero points at z = 0 and z = 2 (K_{cont} and K_{cont,z = 2}), and the emission line contribution (K_{em}). 

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To estimate the Kcorrection of the Gaia photometric Gband, we computed the correlation between the Gband spectral transmission function (in terms of wavelengths) with the QSO synthetic spectrum derived from SDSS QSO spectra by Vanden Berk et al. (2001). The synthetic spectrum ranging from 3800 to 9200 Å could not cover the entire wavelength range of the G filter (3210−11020 Å in the observer comoving reference frame). Because the synthetic spectrum was well fitted by a continuum spectrum with spectral index α_{ν} = −0.46 between the Lyα and Hβ lines, and α_{ν} = −1.58 for wavelengths longer than the Hβ line (Vanden Berk et al. 2001), we extrapolated the synthetic spectrum toward longer wavelengths assuming a continuum spectrum with spectral index α_{ν} = −1.58.
We computed the contribution of the continuum to the Kcorrection assuming a break in the spectral index, considering the zero points of the KCorrection at z = 0 and z = 2, K_{cont} and K_{cont,z = 2}, respectively. The behavior of K_{cont,z = 2} as a function of redshift is displayed in Fig. A.1.
Finally, the contribution to the apparent magnitude of the emission lines K_{em} is given by K_{em} = K−K_{cont,z = 2}. The computed Kcorrection is shown as a function of redshift in Fig. A.1.
To convert the absolute magnitudes M_{i} from the SDSS iband to the Gaia Gband absolute magnitude M, we used the Kcorrection for the SDSS iband given in Richards et al. (2006), the Gband Kcorrection previously described, and calculated an average apparent magnitude transformation between the Gaia Gmagnitude and SDSSi magnitude as follows: (A.2)where is the synthetic QSO spectrum from Vanden Berk et al. (2001), and are the spectral transmission of the G and i bands, considering both magnitudes in the ABmagnitude system. The behavior of as a function of redshift is shown in Fig. A.2.
Fig. A.2 Average color transformation as a function of QSO redshift assuming the source spectral type to be that of Vanden Berk et al. (2001) and both magnitude systems to be ABmagnitudes. 

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The absolute magnitude average transformation is given by (A.3)where and are the Kcorrections in the G and SDSS iband, respectively.
Appendix B: ω_{zd} dependence on the deflector type
In this appendix, we demonstrate that ω_{zd} is independent of the deflector type under the assumption of constant deflector comoving density and a perfect instrument. For simplicity, we consider the case of deflectors modeled by an SIS mass distribution.
The expression of τ_{SIS} in Eq. (8) can be further developed by inserting the expression of which, using the definition of θ_{E} in Eq. (2) and that of Σ_{SIS} in Eq. (3), may be written as (B.1)Inserting the latter expression into Eq. (8) leads to (B.2)where we defined . The coefficient in brackets includes all the dependence on the parameters describing the VDFs (Φ_{∗},σ_{∗},α,β), which may refer to either early or latetype galaxies. In our development, the deflector comoving density is assumed constant, therefore the VDF parameters do not depend on the redshift.
We may obtain an expression for by differentiating the former expression of τ_{SIS} with respect to the deflector redshift z. The result is (B.3)For a perfect instrument, τ_{SIS} and depend in an identical way on the VDF parameters, shown in the first factor of Eqs. (B.2) and (B.3). Therefore, when calculating the normalized redshift
distribution of the deflectors by inserting Eqs. (B.2) and (B.3) into the definition of ω_{zd} (Eq. (18)) and using the definition of ⟨τ⟩ (Eq. (16)), the factors containing the VDF parameters cancel each other out as long as we assume no redshift or absolute magnitude dependence of the VDF parameters. For a perfect instrument, ω_{zd} is thus identical for early and latetype galaxy deflectors because it does no longer depend on the parameters defining the deflector VDF (Φ_{∗},σ_{∗},α, and β).
However, when the finite angular resolution of the survey is taken into account, the integration area for the lensing cross section S_{y} is now a function of the ratio θ_{min}/θ_{∗}, with . This dependence on σ_{∗} leads to different results for the two deflector types because this parameter differs for the early and latetype galaxy VDF.
All Tables
Comparison of the mean lensing optical depth and the expected number of detected multiply imaged quasars for different values of the survey angular resolution corresponding to a perfect survey (θ_{min} = 0′′), Gaia (θ_{min} = 0.2′′), and typical groundbased observations (θ_{min} = 0.6′′).
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Fig. 1 Luminosity function as a function of the absolute magnitude M_{i} in the SDSS iband for different redshift bins. We have included the LF determined by Richards et al. (2005) and PalanqueDelabrouille et al. (2013) (converted to M_{i} magnitude). The different fitted models correspond to no contraints on the redshift behavior of M_{∗} and Φ_{∗} (dashed light gray line), to an evolution model for Φ_{∗} and M_{∗} is fitted freely (light gray line), and to M_{∗} and Φ_{∗} both constrained by an evolution model (continuous black line). The latter is the final evolution model chosen for the simulations, see main text for the full description. 

Open with DEXTER  
In the text 
Fig. 2 Behavior of the LF Φ_{∗} and M_{∗} parameters as a function of the QSO redshift. The dark continuous line shows the fit model used in our simulations. The light gray markers show the best fit parameters for Φ_{∗} and M_{∗} when fitting the LF separately in each redshift bin, without any evolution model constraints. When constraining the evolution of the characteristic density, the best parameters are found to be log Φ_{∗ ,lowz} = −5.85, α_{Φ∗} = −0.77 ± 0.31, and z_{ref} = 2.09 ± 0.28. 

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In the text 
Fig. 3 Left: joint probability density for the Gaia sources, derived using the LF evolution model described in the text. Right: DNCF as a function of the Gband magnitude. We use a combined observational sample of the SDSSDR3 and 2QZ/6QZ for magnitudes brighter (fainter) than i ~ 19, converted to Gband magnitudes and thus assuming the DNCF shape in Gband to be similar. We also show the fit used to estimate the DNCF. 

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In the text 
Fig. 4 Left: average lensing optical depth as a function of the survey angular resolution θ_{min} for early and latetype galaxy deflectors. We modeled the deflectors as SIS (⟨τ_{SIS}⟩continuous black line) and SIE mass distributions (⟨τ_{SIE}⟩continuous gray line). In the latter case, we also show the average optical depth with detection of two lensed images (⟨τ_{SIE,2}⟩ – dashed gray line). Right: fraction of lensed sources detected with three (light gray) or four (dark gray) lensed images as a function of θ_{min}. The results for the case of early and latetype galaxy deflectors are shown with continuous and dashdotted lines, respectively. In both panels we indicate the typical value of θ_{min} for the Gaia survey and typical seeinglimited groundbased observations. 

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In the text 
Fig. 5 Left: expected normalized distributions ω as a function of the redshift of the lensed sources (z_{s}) and of the deflectors (z_{d}). For comparison, we also include the normalized redshift distribution of the sources (ω_{QSO}). The simulations were made assuming θ_{min} = 0.2′′. We indicate the median value of each distribution (, and ). Right: normalized distribution of the lensed sources as a function of the apparent magnitude. For comparison, we show the DNCF as a function of the magnitude of all the detected sources, normalized by N_{QSO}/ Ω_{Gaia}. 

Open with DEXTER  
In the text 
Fig. 6 Left: normalized redshift distribution of the deflectors and its cumulative representation for three different values of the angular resolution θ_{min} = 0,0.2 and 0.6′′. Right: cumulative distribution τ^{1}dτ/dz as a function of the lens redshift. We show the case for two different source redshifts (z_{s} = 2 and 4) and three different values of the angular resolution parameter (θ_{min} = 0,0.6 and 1′′), modeling the deflectors by means of the SIS and SIE models. 

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In the text 
Fig. 7 Left: average optical depth ⟨τ⟩ (considering all deflector types) as a function of cosmological matter density parameter Ω_{m}, modeling the deflectors with the SIS and SIE models. Right: normalized redshift distributions ω_{zs} and ω_{zd} of the lensed sources and of the deflectors (for the earlytype galaxy population) for different values of Ω_{m} = 0,0.3 and 1. All simulations were produced for θ_{min} = 0.2′′. 

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In the text 
Fig. 8 Effect of the cosmological matter density parameter Ω_{m} on the normalized Einstein angular radius distribution of the lensed sources. We considered the deflectors to be modeled by SIS deflectors and an angular resolution θ_{min} = 0.2′′ corresponding to the Gaia survey. θ_{E} is expressed in arcsecond. 

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In the text 
Fig. A.1 Kcorrection in the Gaia photometric Gband as a function of redshift. We represent the total Kcorrection (K) with zero point of the continuum contribution at z = 2, the continuum contributions with zero points at z = 0 and z = 2 (K_{cont} and K_{cont,z = 2}), and the emission line contribution (K_{em}). 

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In the text 
Fig. A.2 Average color transformation as a function of QSO redshift assuming the source spectral type to be that of Vanden Berk et al. (2001) and both magnitude systems to be ABmagnitudes. 

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In the text 
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