aa r X i v : . [ a s t r o - ph ] M a y Is there a quad problem among optical gravitationallenses?
Masamune Oguri
Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 2575Sand Hill Road, Menlo Park, CA 94025, USAE-mail: [email protected]
Abstract.
Most of optical gravitational lenses recently discovered in the SloanDigital Sky Survey Quasar Lens Search (SQLS) have two-images rather than four-images, in marked contrast to radio lenses for which the fraction of four-image lenses(quad fraction) is quite high. We revisit the quad fraction among optical lenses bytaking the selection function of the SQLS into account. We find that the currentobserved quad fraction in the SQLS is indeed lower than, but consistent with, theprediction of our theoretical model. The low quad fraction among optical lenses,together with the high quad fraction among radio lenses, implies that the quasar opticalluminosity function has a relatively shallow faint end slope. s there a quad problem among optical gravitational lenses?
1. Introduction
Strongly lensed multiple quasars have been known to provide an unique probe of ouruniverse. In particular, the point-source nature of quasars allows a simple statisticalstudy from image multiplicities: Statistics of the number of multiple images provideconstraints on the ellipticity and density profile of lens objects as well as the faint endluminosity function of source quasars [1, 2, 3, 4, 5, 6, 7, 8, 9].The statistics of image multiplicities have been done mainly using radio lenses.[5] adopted a radio lens sample of the Cosmic Lens All-Sky Survey (CLASS) [10, 11]to show that the fraction of four-image (quadruple) lenses is significantly higher thanexpected from a standard mass model of elliptical galaxies. [6] showed that the fractionof quadruple lenses in a statistical subsample of the CLASS is marginally consistentwith what we expect from the observed galaxy population, but it still requires relativelylarge galaxy ellipticities.Recent large-scale optical surveys allow us to conduct complementary statisticsusing optical gravitational lenses. In particular, a large sample of quasars discovered inthe Sloan Digital Sky Survey (SDSS) [12] is quite useful for a strong lens survey: Indeed,the SDSS Quasar Lens Search (SQLS) [13] has already discovered approximately 20new strongly lensed quasars (see, e.g., [14] and references therein), becoming the largeststatistical sample of strongly lensed quasars. Interestingly, the fraction of four-imagelenses (quad fraction) in the SQLS appears to be significantly lower than the CLASS.Only a few lenses among ∼
20 new SQLS lensed quasars are quadruple lenses, whereasnearly half of CLASS lenses were four (or more) image systems.In this paper, we revisit the quad fraction among optical gravitational lenses. Weadjust the selection function to that of the SQLS and make a comprehensive prediction ofthe fraction of quadruple lenses. A particular emphasis is paid to whether the current lowquad fraction in the SQLS is consistent with the observed galaxy properties. Throughoutthe paper we adopt Λ-dominated cosmology with the matter density Ω M = 0 . Λ = 0 .
2. Calculation
We assume that the mass distribution of galaxies can be approximated by an SingularIsothermal Ellipsiod (SIE). The scaled surface mass density of an SIE is given by κ ( x, y ) = θ E λ ( e )2 " − e (1 − e ) x + y / , (1)where e denotes the ellipticity. The Einstein radius θ E (for e = 0) is related with thegalaxy velocity dispersion σ by θ E = 4 π (cid:18) σc (cid:19) D ls D os , (2) s there a quad problem among optical gravitational lenses? D ls and D os being the angular diameter distance from lens to source and fromobserver to source, respectively. The normalization factor λ ( e ) basically depends onthe shape and viewing angle of galaxies: In this paper we assume that there are equalnumber of oblate and prolate galaxies and adopt the average of the two normalizations(see [6]). We find that with this normalization the Einstein radii are roughly equal fordifferent ellipticities.It is expected that the quad fraction is mainly determined by the ellipticity.Although the external shear also produces the quadrupole moment in lens potentials,the effect is expected to be minor. For instance, the standard strength of external shear(median value of < .
05) can cause notable changes in the quad fraction only for lensgalaxies with e < . lensmodel [15]. The lensing crosssection σ lens is computed by summing up source positions that yield multiple images witha weight of Φ( L/µ ) /µ/ Φ( L ), where Φ( L ) is the luminosity function of source quasars and µ is the magnification factor (see § z = z s becomes dp i dθ = Z z s dz l c dtdz l (1 + z l ) Z dσ dσ lens ,i d ˜ θ dndσ δ ( θ − ˜ θ )Θ( i gal − i qso ) , (3)with dn/dσ being the velocity function of galaxies. The suffix i = 2 or 4 denote thenumber of images.In computing the lensing probability, we need to specify the lens galaxy population.Since strong lensing is mostly caused by early-type galaxies, particularly for stronglenses in the SQLS whose image separations are basically larger than 1 ′′ , we onlyconsider early-type galaxies. For the velocity function, we assume that of early-typegalaxies derived from the SDSS [17, 18]. More important for the quad fraction is thedistribution of ellipticities. We adopt a Gaussian distribution with mean ¯ e = 0 . σ e = 0 .
16, which is consistent with observed ellipticity distributions ofearly-type galaxies [19, 20, 21, 22, 23], as a fiducial distribution. However we also varythe mean ellipticity, ¯ e , to see how the quad fraction depends on the ellipticity. The quasar luminosity function is another important element to make an accurateprediction of the quad fraction. As a fiducial luminosity function, we adopt thatconstrained from the combination of the SDSS and 2dF [24]:Φ( M g ) = Φ ∗ . − β h )( M g − M ∗ g ) + 10 . − β l )( M g − M ∗ g ) , (4) s there a quad problem among optical gravitational lenses? Table 1.
A current statistical sample of lensed quasars in the SQLS. N img indicatethe number of quasar images.Name N img i PSF
Ref.SDSS J0246 − − where a pure luminosity evolution with M ∗ g ( z ) = M ∗ g (0) − . k z + k z ) (5)is assumed. The parameters are β h = 3 . β l = 1 .
45, Φ ∗ = 1 . × − Mpc − mag − , M ∗ g (0) = − . k = 1 .
39, and k = − .
29. We convert rest-frame g -band magnitudesto observed i -band magnitudes using K-correction derived in [25].The selection function of the SQLS was studied in detail in [13]. Since the statisticalsample of lensed quasars is constructed from quasars with i < . . < z < . θ > ′′ the completeness is almost unity, but there is a small difference of completenessbetween double and quad lenses: To take this into account we include completeness φ i ( θ ) in our calculation. In summary, we compute the numbers of double and quadlenses as n i = Z . . dz s Z i
The fraction of quadruple lenses p Q as a function of i -band limitingmagnitude i lim . Here we consider lensed quasars with redshifts 0 . < z < .
2, fluxratios f i > − . , image separations 1 ′′ < θ < ′′ , and lens galaxies fainter thanthe quasar components i gal − i qso >
0. Dotted line indicate the limiting magnitude ofSDSS quasars, i = 19 .
1. Left: From lower to upper solid lines, the faint end luminosityfunction of quasars β l is changed from 1 .
05 to 1 .
85. The mean ellipticity ¯ e is fixed to0.3. Right: From lower to upper solid lines, the mean ellipticity ¯ e is changed from 0 . .
5. The slope β l is fixed to 1.45. is still to be finalized, we use these lenses to make a tentative comparison with thetheoretical expectation. To make a fair comparison with theory, we select a subsampleof lenses by choosing lenses with redshifts 0 . < z < .
2, magnitudes i < . i -band flux ratios (for doubles) f i > − . , image separations 1 ′′ < θ < ′′ , and lensgalaxies fainter than the quasar components i gal − i qso >
0. Currently we have 13 lensedquasars that meet these conditions, which are summarized in Table 1. Among these 13lenses only two are quadruple lenses, thus the observed quad fraction for the flux limit i lim = 19 . p Q = 2 / ≃ .
3. Result
Before comparing our calculation with the observed quad fraction, we see how itdepends on parameters. Among others, the most important parameter is the ellipticity.Another important element that determines the quad fraction is the shape of the quasarluminosity function. In particular the faint end slope β l still contains large errors becausecurrent large-scale surveys are not deep enough to fully explore the faint end luminosityfunction. For instance, [37] and [38] adopted the 2dF quasar sample to derive the faintend slopes of β l = 1 .
58 and 1 .
09, respectively. A survey of faint quasars conducted by [39] s there a quad problem among optical gravitational lenses? Figure 2.
The quad fraction in our fiducial model (¯ e = 0 . β l = 1 .
45; shown bya solid line) is compared with observed fractions in the SQLS (filled triangles witherrorbars). The errors indicate 68% error estimated assuming the Poisson distributionfor the numbers of double and quad lenses. See table 1 for the lens sample we use.Note that the data points are not independent but rather correlated in the sense thatlenses used to plot at each i lim are included in computing data points at larger i lim aswell. suggests that the faint end slope could be β l = 1 .
25, shallower than our fiducial value.Other uncertainties, such as cosmological parameters, the velocity function of galaxies,and the number of source quasars, affect the number of double and quad lenses roughlysimilarly, thus they hardly change the fraction of quad lenses. We find that the effectof changing the prolate/oblate fraction is not large, affecting the quad fraction only bya few percent. Therefore, in figure 1 we plot the quad fraction as a function of thelimiting magnitude i lim changing these two important parameters, the mean ellipticity ¯ e and the faint end slope β l . First, the quad fraction decreases as the limiting magnitudeincreases. Larger magnifications of quads than doubles indicate that the quad fractionis a strong function of magnification bias such that larger magnification bias results inlarger quad fraction, which explain the decrease of the quad fraction with increasing i lim . As expected, the quad fraction is quite sensitive to the ellipticity and the faint endslope of the quasar luminosity function.Next we compare the quad fraction in our fiducial model with the observed fractionin the SQLS. Figure 2 shows both the theoretical and observed quad fractions as afunction of the limiting magnitude. We find that the observed quad fraction is indeedlower than the theoretical prediction. For instance, at i lim = 19 . p Q = 0 .
273 that is larger than the observation, p Q = 0 . s there a quad problem among optical gravitational lenses? Figure 3.
Probability of our model producing quad lenses equal or fewer than N quad in a sample of 13 lenses, computed from our model prediction of the quad fraction for i lim = 19 . p Q = 0 . e = 0 . p Q = 0 . N quad = 2, is indicated by a verticaldotted line. In figure 3 we plot the probability that our theoretical model produces ≤ N quad quad lenses in a sample of 13 lenses. Note that in observation there are N quad = 2 quadlenses (see table 1). In our fiducial model the probability is ≃ .
27, which is low butacceptable. On the other hand, if we increase the mean ellipticity to ¯ e = 0 .
4, whichis roughly the best-fit value for the observed quad fraction in the CLASS (see [6]), theprobability reduces to ≃ .
05. Therefore with such large-ellipticity model it is difficultto account for the low quad fraction observed in the SQLS.Finally we check the dependence of the likelihood for N quad ≤ β l and the mean ellipticity ¯ e in figure 4. As expectedfrom figure 1, the probability depends sensitively on these parameters. For instance,by decreasing the faint end slope to β l = 1 .
25, which is preferred by a spectroscopicsurvey of faint quasars [39], the probability is increased to ≃ .
35. Changing the meanellipticity to 0 . ≃ .
61, making the observed low quadfraction quite reasonable.
4. Summary and Discussion
In this paper, we have studies the fraction of four-image lenses among opticalgravitational lenses. We have paid a particular emphasis to whether the low quadfraction observed in the SQLS is consistent with the standard theoretical prediction.In order to make a fair comparison, we have taken account of the selection functionand source population in predicting the quad fraction. We find that the observed quad s there a quad problem among optical gravitational lenses? Figure 4.
Probability of our model producing quad lenses equal or fewer than theobserved case, N quad = 2, is plotted as a function of β l ( left ) or ¯ e ( right ). The fiducialvalues are shown by vertical dotted lines. fraction in the SQLS, p Q = 2 / ≃ . p Q = 0 . both optical and radio quadfractions, therefore such models have difficulty in explaining the high quad fractionamong CLASS lenses. For instance, from the CLASS lens sample [6] derived 68% lowerlimit of the mean ellipticity to 0.28 which is marginally consistent with our fiducialmodel, ¯ e = 0 .
3. Therefore, one way to explain both the high quad fraction among radiolenses and low quad fraction among optical lenses is to consider a shallow faint end slopeof the quasar optical luminosity function while keeping the mean ellipticity relativelyhigh.A caveat is that the SQLS is still ongoing and the lens sample is not yet finalized.We should use a final, larger lens sample of the SQLS to draw a more robust conclusionfrom the quad fraction. The final statistical sample is expected to contain roughly twicethe number of lenses we used in this paper, thus the statistical error should be reducedsignificantly.
Acknowledgments
I thank Naohisa Inada and Kyu-Hyun Chae for discussions, and an anonymous refereefor many suggestions. This work was supported in part by the Department of Energy s there a quad problem among optical gravitational lenses?
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