Draft version January 3, 2019 Typeset using LATEX twocolumn style in AASTeX61 MOST LENSED QUASARS AT Z > 6 ARE MISSED BY CURRENT SURVEYS Fabio Pacucci1 and Abraham Loeb2 1Yale University, Department of Physics, New Haven, CT 06511, USA 2Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA ABSTRACT The discovery of the first strongly lensed (µ ≈ 50) quasar at z > 6 (J0439+1634) represents a breakthrough in our understanding of the early Universe. We derive the theoretical consequences of the new discovery. We predict that the observed population of z > 6 quasars should contain many sources with magnifications µ . 10 and with image separations below the resolution threshold. Additionally, current selection criteria could have missed a substantial population of lensed z > 6 quasars, due to the contamination of the drop-out photometric bands by lens galaxies. We argue that this predicted population of lensed z > 6 quasars would be misclassified and mixed up with low-z galaxies. We quantify the fraction of undetected quasars as a function of the slope of the bright end of the quasar luminosity function, β. For β . 3:6, we predict that the undetected lensed quasars could reach half of the population, whereas for β & 4:5 the vast majority of the z > 6 quasar population is lensed and still undetected. This would significantly affect the z > 6 quasar luminosity function and inferred black hole mass distributions, with profound implications for the ultraviolet, X-ray, and infrared cosmic backgrounds and the growth of early quasars. Keywords: gravitational lensing: strong | galaxies: active | galaxies: luminosity function, mass function | quasars: supermassive black holes | early universe | dark ages, reionization, first stars arXiv:1810.12302v2 [astro-ph.GA] 2 Jan 2019 Corresponding author: Fabio Pacucci [email protected] 2 Pacucci & Loeb 1. INTRODUCTION 2. THEORETICAL FRAMEWORK The discovery of the first strongly lensed quasar at We employ the formalism developed in Pei(1995) to z > 6 (Fan et al. 2018) marks a breakthrough nearly analytically compute the magnification probability dis- two decades after the first theoretical prediction of their tribution, P (µ), due to cosmologically distributed galax- existence (Wyithe & Loeb 2002). Fan et al.(2018) re- ies. We define µ as the total magnification of a source 0 port the detection of J0439+1634, a z = 6:51 quasar at a redshift zs, due to lenses at redshift z . The prob- whose best-fit model predicts three images and a total ability distribution of magnifications larger than µ is magnification µ = 51:3 ± 1:4. This lensed quasar is the Z 1 first example of this class detected during the reioniza- P (> µ) = P (µ0)dµ0 : (1) tion epoch (Barkana & Loeb 2001; Fan et al. 2006). µ Wyithe & Loeb(2002) suggested the possibility of a Following Pei(1993, 1995), we distinguish between the high fraction (up to ∼ 1=3) of lensed sources among magnification µ, relative to a filled beam in a smooth high-z quasars, due to the large lensing optical depths Universe, and the magnification A, relative to an empty reached at z 6. This fraction of the then-known pop- & beam in a clumpy Universe. Calling A¯ the mean magni- ulation of high-z quasars would have had their observed fication, the two variables are related by µ ≡ A=A¯. We flux magnified by factors µ 10. Thus far, none of the & define the moment function Z(sjzs) as ∼ 150 quasars observed at z & 6 (Ba~nadoset al. 2018) have shown evidence for lensing. Z zs Z 1 As suggested by Fan et al.(2018), this lack of lensed 0 0 Z(sjzs) = dz dAρ(A; z jzs)× sources could be accounted for by a selection bias. The 0 1 (2) selection of z & 6 quasars requires a non-detection (or × (As − 1 − 0:4 ln (10)A + s) : at least a strong drop) in the drop-out band at shorter 0 wavelengths than the Lyman break (912 A).˚ In their The quantity ρ(A; z jzs) is the mean number of lenses in detection paper, Fan et al.(2018) point out that the the range (z0, z0 + dz0) with magnifications in the range presence of a lens galaxy along the line of sight of the (A, A + dA) for a given source at a redshift zs. The lensed quasar injects flux into the drop-out bands. This probability distribution function for µ reads effect is very relevant for partly resolved images, a likely Z +1 −1 occurrence for high-z sources. This can thus constitute a P (µ) = µ ds exp [−2πis ln µ + Z(2πisjzs)] : relevant issue in the selection of z & 6 lensed quasars. As −∞ the drop-out bands also contain transmission windows (3) of the intergalactic medium (IGM), Fan et al.(2018) With some analytical adjustments, this integral can be caution that great care is needed in separating the con- efficiently computed via a fast Fourier transform algo- tinuum of the lens galaxy from the transmission spikes rithm. of the IGM. Assuming that the population of lenses is composed of This new discovery has several key implications. First, galaxies with flat rotation curves modeled as a truncated it suggests the likely existence of a population of more singular isothermal sphere, their surface mass density modestly lensed quasars that are thus far undetected profile can be written as (Fan et al. 2018). Second, if this population of unde- −2 σ2 R tected quasars does exist, it could potentially impact in Σ(R) = v 1 + ; (4) 2GR R a very significant way the luminosity and mass functions T of the earliest populations of supermassive black holes where σv is the line-of-sight velocity dispersion, R is the (BHs). In this regard, Wyithe & Loeb(2002) pointed projected radius, RT is the radius containing half of the out that the inclusion of a magnification bias in the high- projected mass, and G is the gravitational constant. The 2 z quasar surveys could have a major effect on the abun- total mass is finite and equal to MT = πσvRT =G. For 13 dance of massive halos (Mh & 10 M ), for which the such a population of lenses, calling δ(x) the Dirac delta mass function is very steep. function In this Letter we derive the theoretical consequences Z 1 0 0 of the discovery of the first strongly lensed quasar in ρ(A; z jzs) = 2τ(z jzs) dffδ[A − A(f)] : (5) the epoch of reionization (Fan et al. 2018). Throughout 0 the Letter we use the latest values of the cosmological 0 The coefficient τ(z jzs) is the strong lensing optical parameters from the Planck Collaboration et al.(2018). depth between source and lens galaxy at z0 (see Pei 0:5 1995). The factor f is defined as f = l=(acrRT ) , Most Lensed Quasars at z > 6 are Missed by Current Surveys 3 where l is the unperturbed impact parameter, and acr is 0.0 the critical impact parameter for double images. In Eq. 5, A(f) = A+(f) for f > 1=r and A(f) = A+(f)+A−(f) 0.5 for f ≤ 1=r, with P(µ > 2, zs > 6) = 0. 20 1.0 2 1 f ± fr + 2 ) A±(f) = ± 1 ; (6) µ 2 f[(f ± r)2 + 4]1=2 > P(µ > 2, z > 6) = 0. 04 ( 1.5 s P g o 1=2 L RT cH0 2.0 Observed frequency of r = 2 0 : (7) lensed z > 6 quasars 4πσvD(zs; z ) P( > µ): β = 3. 6 0 2.5 Here, D(zs; z ) is the angular diameter distance between P( > µ): β = 2. 8 0 source and lens at z , and c is the speed of light. The size J0439 + 1634 : zs = 6. 51, µ = 51. 3 1. 4 −1=2 ± parameter r has dimensions of length and is usually 3.0 0.0 0.5 1.0 1.5 2.0 expressed as a function of the dimensionless parameter Log µ 2 0 F = 3ΩG=[2r D(zs; z )], with ΩG the density parameter of galaxies. Following Pei(1995), we assume F ∼ 0:05 Figure 1. The probability distribution function P (> µ, zs = and we check that our results are unchanged throughout 6:51) is shown for β = 2:8 and β = 3:6, and for magnification the range 0:01 < F < 0:1, which covers the full domain values between µ = 1 and 100. The dashed vertical line indi- cates the magnification factor for the lensed quasar reported of interest. by Fan et al.(2018), with the 1 σ uncertainty level shown as a shaded region. Note that P (µ > 50) = Pobs for β ≈ 3:2. 2.1. Distribution of lens galaxies The theoretical framework described thus far to make range includes the population of lens galaxies that con- predictions for the lensing probability P (> µ) has to be tribute the most to the lensing optical depth. We model supplemented with a realistic cosmological distribution the redshift evolution of the stellar velocity dispersion of foreground (z 6) galaxies, assumed to be early type γ . as σv(z) / (1 + z) . Following Beifiori et al.(2014), we (E/S0) and distributed uniformly in space. The ultra- use the value γ = 0:18 ± 0:06, indicating a mild evolu- violet (UV) luminosity function for galaxies is modeled tion. This value is consistent with other studies (e.g., as a simple Schechter function (Schechter 1976): Mason et al. 2015), but smaller than what found by van αg de Sande et al.(2013). As discussed in Mason et al. Φ? L L Φ(L) = exp − ; (8) (2015), measurements of velocity dispersions at z & 0:5 L? L? L? are very difficult, and large uncertainties in lens models result from a lack of knowledge of how the dark matter where L? is the break luminosity, Φ? is the number den- contained in galaxies (traced by σv) evolves with red- sity of galaxies of luminosity L?, and αg is the slope of the faint end.