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Magnification of light from many distant quasars by gravitational lenses



Exceptionally bright quasars with redshifts up to z = 6.28 have recently been discovered1. 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 Universe2,3, as it requires their formation within one billion years of the Big Bang. Here we show that up to one-third 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.


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 known1 (with z 5.8; SDSS 1044-0125 was later found4 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 × 109M, where M is the solar mass, implying3 that the mass of their host galaxies is 1013M. The evolution of the space density of halos of a given mass is described by the Press–Schechter5 mass function. Such massive hosts lie on the steep exponential tail of this mass function, so that a correction to the inferred black-hole 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 mass3, 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 lower-mass 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 implications6,7 with respect to whether the cosmic neutral hydrogen, a cold remnant from the Big Bang, was re-ionized by star light or by quasars8,9.

Figure 1: Enhancement in the co-moving space density of host galaxies for quasars at z = 6 as a function of the magnification due to lensing, μ.

Ignoring lensing, the Eddington limit implies3 a minimum black hole mass of about 3 × 109M for the SDSS quasars at z ≈ 6. The inclusion of lensing reduces the implied (minimum) black hole mass by a factor of μ. Given some fixed efficiency for assembling gas into a central black hole within a galaxy, the implied mass of the host galaxy is also lowered by μ after lensing is included. Plotted is Δ(μ) = {dn/d(log [M/μ])}/{dn/d(log[M])} where dn / d(log M) is the Press–Schechter5 co-moving density of galaxy halos with mass M per logarithmic interval in M. We show curves for three possible halo masses of quasar hosts, Mhalo = 1012M, 1013M, and 1014M. As seen, the enhancement in the implied abundance of quasar hosts increases with increasing halo mass. The above three choices for halo masses are all conservatively lower than inferred for the same black hole mass in the local universe30. Throughout this work we assume the standard cosmological parameters Ωm = 0.35, ΩΛ = 0.65, Ωb = 0.05, H0 = 65 km s-1 Mpc, n = 1 and fluctuation amplitude σ8 = 0.8.

For the lensing calculation we must specify the quasar luminosity function (number per co-moving volume per unit luminosity). At z 3 this is well described by a double power law10, with all redshift dependence in the evolution8 of the break luminosity L:

We find that an intrinsic luminosity function having slopes at the faint and bright ends of βl = 1.64 and βh = 3.43, and parameters L(0) = 1.5 × 1011L, where L is the solar luminosity, αq = -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 B-magnitude MB < -26 at z ≈ 4.3 (measured by SDSS11) and MB < -27.6 at z ≈ 6.0. The parameter αq is the slope assumed for the typical quasar continuum L(ν)Lαq. We use the integral luminosity function N(>Llim,z) = ∫LlimdLφ(L,z), where Llim is the luminosity of a quasar at redshift z corresponding to an apparent magnitude zlim*. Llim was determined from zlim* = 20.2 using a luminosity distance and a k-correction computed from a model quasar spectrum including the mean absorption by the intergalactic medium12.

Gravitational lensing is expected to be highly probable for very luminous quasars13. 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 τmult as the probability that a random quasar selected in the source plane will be multiply imaged14, and FMI to be the magnification-biased probability that an observed quasar will be multiply imaged. Surveys for quasars at z < 3 have limiting magnitudes fainter than the break magnitude (mB ≈ 19). Whereas τmult ≈ 0.002 at z ≈ 2, a survey for quasars to a limit Llim will find a number of lensed sources that is larger by a bias factor of N(<Llim/μ,z)/N(<Llim,z). At z = 2, Llim(z)L(z) and for βl = 1.6 the magnification bias factor is about 2.3, resulting in FMI ≈ 0.005. At z = 6, τmult ≈ 0.008 is significantly higher15 than at z = 2. Furthermore, the limiting magnitude of the z 5.8 survey is significantly brighter than the break magnitude and so βh = 3.4 and the bias factor rises to approximately 28. Under these circumstances, FMI is about 0.22. These simple arguments are consistent with previous estimates16 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 dPsing/dμ and dPmult/dμtot for the magnification μ of randomly positioned singly imaged sources, and for the sum of magnifications μtot of randomly positioned multiply imaged sources due to gravitational lensing by foreground galaxies. We assume that the lens galaxies have a constant co-moving density (as the lensing rate for an evolving (Press–Schechter) population of lenses differs only by 10% from this case16) and are primarily early-type (E/S0) SIS galaxies17 whose population is described by a Schechter function with parameters18 n = 0.27 × 10-2 Mpc-3 and αs = -0.5. We assume the Faber–Jackson relation (Lg/Lg) = (σg/σ)4 where σg 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 dispersion17. We ignore dust extinction by the lens galaxy, which should lower17 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 Lg from the 2dF early-type galaxy luminosity function18, the Faber–Jackson relation, colour transformations with a k-correction19,20, and the evolution of the rest-frame B-band mass-to-light ratio21.

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 stars22 equals 0.005. At z > 0 the total mass density in stars is assumed to be proportional to the cumulative star-formation history23,24. The parameters of the de Vaucouleurs profiles are taken from a study of the fundamental plane25, the microlens mass is 0.1M and the source size 1015 cm (corresponding to ten Schwarzschild radii of a 3 × 108M black hole). The probabilities dPsing//dμ and dPmult//dμtot closely resemble the standard form for the SIS and were computed26 by combining numerical magnification maps with the distribution of microlensing optical depth and shear along lensed lines of sight, and the distribution [τmult{dPmult/dμ} + (1 - τmult){dPsing/dμ}] was normalized to have a unit mean.

The fraction of sources which are multiply imaged due to gravitational lensing is

where τmult = 0.0059. We find a value of FMI ≈ 0.30 for the colour-selected, flux-limited 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 FMI ≈ 0.01 at z ≈ 2 for mB < 20. These calculations do not include selection effects for flux ratios or image separations which may serve to lower FMI, but are specific to follow up observations.

A subarcsecond-resolution K-band image has been obtained for the z = 5.80 quasar1, and it was found to be an unresolved point source. To our knowledge, this is the only z 5.8 quasar for which subarcsecond-resolution optical imaging is currently available. However, a program to image these quasars with the Hubble Space Telescope that will determine the multiple-image fraction is expected to begin within the coming year (X. Fan, personal communication). We note, however, that single-image quasars might still be magnified by a factor of about 2. Recent Chandra observations27 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 Llim at redshift z:

In Fig. 2 we show the probability that the magnification of a quasar is higher than μobs, 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(μobs) ≈ 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.

Figure 2: The probability of observing a magnification larger than μobs for a quasar at a redshift z = 6 in a sample with a magnitude limit z* < 20.2.

The distribution is highly skewed, having a median of med(μobs) = 1.2 and a mean of 〈μobs〉 = 24.0. The multiple image fraction is FMI = 0.30. We have also computed the a posteriori values of FMI and 〈μ〉 for specific quasars. For SDSS 0836-0054 (z = 5.82), SDSS 1306-0356 (z = 5.99) and SDSS 1030-0524 (z = 6.28) we find FMI = 0.40, 0.32 and 0.31, and 〈μ〉 = 50, 25 and 23, respectively. Although Fan et al.1 find that βh = 3.43 is consistent with the luminosities of the z 5.8 quasars, it is possible that the luminosity function at high redshift is not as steep as βh = 3.43. If so, the magnification bias will not be as large and the mean magnification and the fraction of quasars that are multiply imaged will be lower. We have recomputed the lens statistics assuming that the bright end slope is significantly flatter at high redshift. Assuming that βl = 1.64 at all redshifts and βh = 3.43 for z < 3 but βh = 2.58 for z > 3 (the value found11 for quasars at z ≈ 4.3) we inferred similar parameters to describe the observed luminosity function as before (L(0) = 1.5 × 1011L,z = 1.6, ξ = 3.3, ζ = 2.65). Remarkably, the multiple image fraction is still nearly 0.1 in this case. Additional uncertainty in the calculation arises from the choice of lens model. The value τmult is proportional to nσ4. The dependence of FMI on τmult is complex (see equation (3)); however, large magnification biases result in a relation that is less sensitive than linear. For example, reducing the value of τmult by a factor of 1.5 results in FMI = 0.22 if βh = 3.43 and FMI = 0.043 if βh = 2.58.

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 MB < -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 Llim/μ and has its luminosity, L, overestimated by μ. The factor RLD by which the luminosity density of quasars brighter than Llim is overestimated because of lensing is therefore

We find RLD ≈ 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 blue-ward of the Lyα resonance. The resulting Gunn–Peterson28 trough has been observed in the spectrum of the highest redshift quasar7,29, 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 re-ionization.

The high source redshifts imply image separations that are slightly larger than usual16, about 1–2″. In addition, the lenses are likely to be found at higher redshift16 owing in part to the higher source redshifts but also because bright (low-z) lenses are excluded. Lensing of high-redshift quasars allows measurement of the masses of the lens galaxies21 at redshifts higher than is currently possible. Furthermore, quasar microlensing should be common over a ten-year base line26. 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 formation3.


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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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The authors declare that they have no competing financial interests.

Correspondence to Abraham Loeb.

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