Abstract
Exceptionally bright quasars with redshifts up to z = 6.28 have recently been discovered^{1}. Quasars are thought to be powered by the accretion of gas onto supermassive black holes at the centres of galaxies. Their maximum (Eddington) luminosity depends on the mass of the black hole, and the brighter quasars are inferred to have black holes with masses of more than a few billion solar masses. The existence of such massive black holes poses a challenge to models for the formation of structures in the early Universe^{2,3}, as it requires their formation within one billion years of the Big Bang. Here we show that up to onethird of known quasars with z ≈ 6 will have had their observed flux magnified by a factor of ten or more, as a consequence of gravitational lensing by galaxies along the line of sight. The inferred abundance of quasar host galaxies, as well as the luminosity density provided by the quasars, has therefore been substantially overestimated.
Main
Gravitational lensing leads to a magnification by a factor μ of the apparent source luminosity. Let us first consider the implications of lensing for the early existence of massive quasar systems within structure formation theory. The four highest redshift quasars known^{1} (with z ≳ 5.8; SDSS 10440125 was later found^{4} to have z = 5.73), were selected in the SDSS (Sloan Digital Sky Survey) photometric system to have magnitudes z* < 20.2 and colours i*  z* > 2.2. The masses of the central black holes powering these quasars are estimated to be 3 × 10^{9}M_{⊙}, where M_{⊙} is the solar mass, implying^{3} that the mass of their host galaxies is ≳ 10^{13}M_{⊙}. The evolution of the space density of halos of a given mass is described by the Press–Schechter^{5} mass function. Such massive hosts lie on the steep exponential tail of this mass function, so that a correction to the inferred blackhole mass severely affects the estimated cosmological density of galaxy halos which are sufficiently massive to host the observed quasars.
Using the Eddington luminosity to set a lower limit on the inferred black hole mass^{3}, we find that the magnification due to lensing lowers the minimum black hole mass by a factor of μ. This in turn implies that the black hole can form in a lowermass galaxy. Figure 1 shows the resulting enhancement factor in the space density of galaxy halos that could host the observed SDSS quasars at z = 6. The inclusion of lensing has a very large effect (by orders of magnitude) on the expected abundance of such hosts at early cosmic times. The prominence and duty cycle of such hosts has important implications^{6,7} with respect to whether the cosmic neutral hydrogen, a cold remnant from the Big Bang, was reionized by star light or by quasars^{8,9}.
For the lensing calculation we must specify the quasar luminosity function (number per comoving volume per unit luminosity). At z ≲ 3 this is well described by a double power law^{10}, with all redshift dependence in the evolution^{8} of the break luminosity L_{★}:
We find that an intrinsic luminosity function having slopes at the faint and bright ends of β_{l} = 1.64 and β_{h} = 3.43, and parameters L_{★}(0) = 1.5 × 10^{11}L_{⊙}, where L_{⊙} is the solar luminosity, α_{q} = 0.5, z_{★} = 1.45, ξ = 2.9 and ζ = 2.7 adequately (after consideration of the gravitational lensing described below) describes the luminosity function at z ≲ 3, and the number density of quasars with absolute Bmagnitude M_{B} < 26 at z ≈ 4.3 (measured by SDSS^{11}) and M_{B} < 27.6 at z ≈ 6.0. The parameter α_{q} is the slope assumed for the typical quasar continuum L(ν)∝L^{α}_{q}. We use the integral luminosity function N(>L_{lim},z) = ∫_{Llim}^{∞}dL φ(L,z), where L_{lim} is the luminosity of a quasar at redshift z corresponding to an apparent magnitude z_{lim}^{*}. L_{lim} was determined from z_{lim}^{*} = 20.2 using a luminosity distance and a kcorrection computed from a model quasar spectrum including the mean absorption by the intergalactic medium^{12}.
Gravitational lensing is expected to be highly probable for very luminous quasars^{13}. Consider a fictitious gravitational lens that always produces a magnification of μ = 4 for the sum of multiple images (the average value for a singular isothermal sphere, SIS) but μ = 1 otherwise. We define τ_{mult} as the probability that a random quasar selected in the source plane will be multiply imaged^{14}, and F_{MI} to be the magnificationbiased probability that an observed quasar will be multiply imaged. Surveys for quasars at z < 3 have limiting magnitudes fainter than the break magnitude (m_{B} ≈ 19). Whereas τ_{mult} ≈ 0.002 at z ≈ 2, a survey for quasars to a limit L_{lim} will find a number of lensed sources that is larger by a bias factor of N(<L_{lim}/μ,z)/N(<L_{lim},z). At z = 2, L_{lim}(z)≪L_{★}(z) and for β_{l} = 1.6 the magnification bias factor is about 2.3, resulting in F_{MI} ≈ 0.005. At z = 6, τ_{mult} ≈ 0.008 is significantly higher^{15} than at z = 2. Furthermore, the limiting magnitude of the z ≳ 5.8 survey is significantly brighter than the break magnitude and so β_{h} = 3.4 and the bias factor rises to approximately 28. Under these circumstances, F_{MI} is about 0.22. These simple arguments are consistent with previous estimates^{16} and demonstrate that lensing has a strong effect on observations of the bright SDSS quasars at z ≳ 5.8.
To find the magnification bias more accurately, we have computed the probability distributions dP_{sing}/dμ and dP_{mult}/dμ_{tot} for the magnification μ of randomly positioned singly imaged sources, and for the sum of magnifications μ_{tot} of randomly positioned multiply imaged sources due to gravitational lensing by foreground galaxies. We assume that the lens galaxies have a constant comoving density (as the lensing rate for an evolving (Press–Schechter) population of lenses differs only by ≲10% from this case^{16}) and are primarily earlytype (E/S0) SIS galaxies^{17} whose population is described by a Schechter function with parameters^{18} n_{★} = 0.27 × 10^{2} Mpc^{3} and α_{s} = 0.5. We assume the Faber–Jackson relation (L_{g}/L_{g★}) = (σ_{g}/σ_{★})^{4} where σ_{g} is the velocity dispersion of the lens galaxy, with σ_{★} = 220 km s^{1} and a dark matter velocity dispersion that equals the stellar velocity dispersion^{17}. We ignore dust extinction by the lens galaxy, which should lower^{17} the lensing rate by ≲10%. Potential lens galaxies must not be detectable in the survey data used to select the objects. Galaxies having i* < 22.2 (around 30% of the potential lens population) are not considered part of the lens population. To compute i* for a galaxy having velocity dispersion σ at redshift z, we use L_{g★} from the 2dF earlytype galaxy luminosity function^{18}, the Faber–Jackson relation, colour transformations with a kcorrection^{19,20}, and the evolution of the restframe Bband masstolight ratio^{21}.
Our lens model includes microlensing by the population of galactic stars, which is modelled as a de Vaucouleurs profile of point masses embedded in the overall SIS mass distribution. The surface mass density in stars for galaxies at z = 0 is normalized so that the total cosmological density parameter of stars^{22} equals 0.005. At z > 0 the total mass density in stars is assumed to be proportional to the cumulative starformation history^{23,24}. The parameters of the de Vaucouleurs profiles are taken from a study of the fundamental plane^{25}, the microlens mass is 0.1M_{⊙} and the source size 10^{15} cm (corresponding to ten Schwarzschild radii of a 3 × 10^{8}M_{⊙} black hole). The probabilities dP_{sing}//dμ and dP_{mult}//dμ_{tot} closely resemble the standard form for the SIS and were computed^{26} by combining numerical magnification maps with the distribution of microlensing optical depth and shear along lensed lines of sight, and the distribution [τ_{mult}{dP_{mult}/dμ} + (1  τ_{mult}){dP_{sing}/dμ}] was normalized to have a unit mean.
The fraction of sources which are multiply imaged due to gravitational lensing is
where τ_{mult} = 0.0059. We find a value of F_{MI} ≈ 0.30 for the colourselected, fluxlimited z ≳ 5.8 sample. This value is higher by two orders of magnitude than the lens fraction at low redshifts and demonstrates that lensing must already be considered in a sample with only four objects. For comparison, our model predicts F_{MI} ≈ 0.01 at z ≈ 2 for m_{B} < 20. These calculations do not include selection effects for flux ratios or image separations which may serve to lower F_{MI}, but are specific to follow up observations.
A subarcsecondresolution Kband image has been obtained for the z = 5.80 quasar^{1}, and it was found to be an unresolved point source. To our knowledge, this is the only z ≳ 5.8 quasar for which subarcsecondresolution optical imaging is currently available. However, a program to image these quasars with the Hubble Space Telescope that will determine the multipleimage fraction is expected to begin within the coming year (X. Fan, personal communication). We note, however, that singleimage quasars might still be magnified by a factor of about 2. Recent Chandra observations^{27} of the z = 6.28 quasar show photons detected on the edge of the extraction (1.2″) circle, offering a hint that this quasar may be lensed.
We have computed the distribution of magnifications observed for a sample of quasars brighter than L_{lim} at redshift z:
In Fig. 2 we show the probability that the magnification of a quasar is higher than μ_{obs}, assuming that the quasar belongs to a sample at a redshift z ≈ 6 with the SDSS magnitude limit of z* < 20.2. The distribution is highly skewed; the median magnification is med(μ_{obs}) ≈ 1.2, whereas the mean is as high as 〈μ〉 = 24. Thus, one or more of the z ≳ 5.8 quasars are likely to be highly magnified, whereas most should be magnified at a low level.
The large values of the magnifications and the highly skewed shape of the distributions in Fig. 2 suggest that lensing must alter the observed luminosity function. Indeed, we find that the space density of quasars with M_{B} < 27.6 is increased by 40%, and that the slope is decreased by 0.15. A quasar magnified by μ is detectable to luminosities as low as L_{lim}/μ and has its luminosity, L, overestimated by μ. The factor R_{LD} by which the luminosity density of quasars brighter than L_{lim} is overestimated because of lensing is therefore
We find R_{LD} ≈ 2, implying that naive computation of the quasar luminosity density from the z ≳ 5.8 sample might significantly overestimate its true value. Magnification bias also affects quasars that are not multiply imaged. We therefore predict an enhanced angular correlation on the sky between z≈6 quasars and foreground galaxies.
Traces of neutral hydrogen at high redshift absorb flux just blueward of the Lyα resonance. The resulting Gunn–Peterson^{28} trough has been observed in the spectrum of the highest redshift quasar^{7,29}, and is the primary indicator of the reionization epoch. We find that about 40% of multiple image lens galaxies (i* < 22.2) contribute flux in the Gunn–Peterson trough above the current upper limit. Contamination of the Gunn–Peterson trough in lensed quasar spectra will therefore limit their use as a probe of the epoch of reionization.
The high source redshifts imply image separations that are slightly larger than usual^{16}, about 1–2″. In addition, the lenses are likely to be found at higher redshift^{16} owing in part to the higher source redshifts but also because bright (lowz) lenses are excluded. Lensing of highredshift quasars allows measurement of the masses of the lens galaxies^{21} at redshifts higher than is currently possible. Furthermore, quasar microlensing should be common over a tenyear base line^{26}. This offers the exciting possibility of measuring the source size, and hence black hole mass, which in turn yields the ratio between quasar luminosity and its Eddington value, a quantity that will be useful in constraining models of structure formation^{3}.
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Acknowledgements
We thank E. Turner, J. Winn and R. Barkana for discussions. This work was supported in part by grants from the NSF and NASA. J.S.B.W. is supported by a Hubble Fellowship.
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Correspondence to Abraham Loeb.
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Further reading

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