MICROLENSING-BASED ESTIMATE OF THE MASS FRACTION IN COMPACT OBJECTS IN LENS GALAXIES

The composition of matter in the halos of galaxies is a central problem in astrophysics. During the last 10 years, several observational projects have used gravitational microlensing (Paczynski 1986) to probe the properties of the halos of the Milky Way (MACHO, Alcock et al. 2000; EROS, Tisserand et al. 2007) and M31 (POINT-AGAPE, Calchi Novati et al. 2005; MEGA, de Jong et al. 2006). These experiments are based on the detection of magnification in the light curve of a source induced by an isolated point-like (or binary) object passing near the observer's line of sight. From the successful detection of a number of microlensing events these collaborations have estimated the fraction of the halo mass that is composed of lensing objects, α. However, the reported results disagree. For the Milky Way's halo the measurements of the MACHO collaboration (Alcock et al. 2000) correspond to a halo fraction of 0.08 < α < 0.50 while EROS (Tisserand et al. 2007) obtains α < 0.08. On the other hand, re-analysis of publicly available MACHO light curves (Belokurov et al. 2004) leads to results similar to those reported by EROS (however, see also the counter-report by Griest & Thomas 2005). For M31 the AGAPE (Calchi Novati et al. 2005) collaboration finds a halo fraction in the range 0.2 < α < 0.9, while MEGA (de Jong et al. 2006) finds a limit of α < 0.3.

The method applied to the Milky Way and M31 can be extended to the extragalactic domain by observing the microlensing induced by compact objects in the lens galaxy halo in images of multiply imaged quasars (quasar microlensing; Chang & Refsdal 1979, see also the review by Wambsganss 2006). Interpreting the light curves of QSO 2237+0305, Webster et al. (1991) suggest that the monitoring of microlensing variability can provide a measure of the optical depth in compact objects and in the smooth mass distribution. Lewis & Irwin (1996) proposed a statistical approach to the determination of the mass density in compact objects based on the comparison between the observed and simulated magnification probability distributions. Microlensing can also be measured from a single-epoch snapshot of the anomalous flux ratios induced by this effect between the images of a lensed quasar (Witt et al. 1995; see also Schechter & Wambsganss 2002). Schechter & Wambsganss (2004) explore the practical application of this idea by using a sample of 11 systems with measured flux anomalies. Other quasar microlensing studies of interest for the present study are aimed at the determination of accretion disk sizes (e.g., the studies based in relatively large samples by Poindexter et al. 2007, Morgan et al. 2007 and references therein).

In practice, the study of extragalactic microlensing meets significant obstacles, in particular (e.g., Kochanek 2004) larger timescales for microlensing variability and lack of a baseline for no magnification needed to detect and to quantify microlensing (see, however, the time variability based studies of several individual systems in Morgan et al. 2008 and references therein). In addition, microlensing by an isolated object is not a valid approximation. Microlensing at high optical depth should be modeled (e.g., by simulating magnification maps; see Schneider et al. 1992).

We avoid these obstacles by setting the baseline of no microlensing magnification using the narrow emission lines (NELs) in the spectra of lensed quasar images (Schechter & Wambsganss 2004 follow a similar approach but using theoretical models to define the baseline). It is generally expected that the regions where NELs originate are very large (compared with the continuum source) and are not affected by microlensing (this assumption can also be adopted, to some extent, for low ionization broad emission lines; Kaspi et al. 2000; Abajas et al. 2002). If we define the baseline from emission lines measured in the same wavelength regions as the continua affected by microlensing, we can also remove the extinction and isolate the microlensing effects.

"Intrinsic" flux ratios between the images in the absence of microlensing can be determined from the observation of the mid-infrared and radio-emitting regions of quasars that should also be large enough to average out the effects of microlensing (see Kochanek 2004 and references therein). However, the extinction at mid-infrared and radio wavelengths is lower than the extinction at the wavelengths in which microlensing is usually detected and measured (optical, near-infrared, and X-ray). Consequently, the difference between the mid-infrared (radio) and the optical (X-ray or near-infrared) continuum fluxes will include not only the effects of microlensing but also the effects of extinction. In addition, note that the availability of data at optical wavelengths is considerably greater than at other wavelengths.

Thus, we will use the NEL and continuum flux ratios among the different images of a lensed QSO to estimate the difference of microlensing magnification between the images at a given epoch with certain restrictions that we detail in the following paragraphs.

The flux (in magnitudes) of an emission line observed at wavelength λ of image i of a multiply imaged quasar is equal to the flux of the source, $m_0^{\rm lin} \, \big({\lambda \over 1 \,\, + \ z_s}\big)$, magnified by the lens galaxy (with a Φ_i magnification factor; μ_i = −2.5log Φ_i) and corrected by the extinction of this image caused by the lens galaxy, $A_i\big({\lambda \over 1 \, + \ z_l}\big)$ (see, e.g., Muñoz et al. 2004),

$$\begin{equation} m_i^{\rm lin}(\lambda)= m_0^{\rm lin}\left({\lambda \over 1 +z_s}\right) + \mu _i + A_i\left({\lambda \over 1 +z_l}\right), \end{equation} \tag{ 1 }$$

where z_s and z_l are the redshifts of the source and the lens, respectively.

In the case of the continuum emission, we must also take into account the intrinsic variability of the source combined with the delay in the arrival of the signal, which is different for each image, Δt_i, and the microlensing magnification, which depends on wavelength and time (with a $\phi _i\big[{\lambda \over 1 \, + \ z_l},t \big]$ magnification factor; Δμ_i = −2.5log ϕ_i),

$$\begin{eqnarray} m_i^{\rm con}(\lambda,t)&=& m_0^{\rm con}\left({\lambda \over 1 +z_s},t-\Delta t_i\right) + \mu _i + A_i\left({\lambda \over 1 +z_l}\right) \nonumber\\ &&+ \Delta \mu _i \left({\lambda \over 1 +z_l},t\right). \end{eqnarray} \tag{ 2 }$$

Thus, the difference between continuum and line fluxes cancels the terms corresponding to intrinsic magnification and extinction (μ_i + A_i):

$$\begin{eqnarray} m_i^{\rm con}(\lambda,t) - m_i^{\rm lin}(\lambda)&=& m_0^{\rm con}\! \left({\lambda \over 1 +z_s},t-\Delta t_i\right) - m_0^{\rm lin}\! \left({\lambda \over 1 +z_s}\right)\nonumber\\ && + \Delta \mu _i\left({\lambda \over 1 +z_l},t\right). \end{eqnarray} \tag{ 3 }$$

If we consider a pair of images, 1 and 2, the continuum ratio relative to the zero point defined by the emission line ratio can be written (in magnitudes) as

$$\begin{eqnarray} \Delta m (\lambda,t)&=& (m_1-m_2)^{\rm con}-(m_1-m_2)^{\rm lin}\nonumber\\ &=&{\Delta \mu _1\left({\lambda \over 1 +z_l},t\right)- \Delta \mu _2\left({\lambda \over 1 +z_l},t\right) } \\ &&+\Delta m_0^{\rm con}\left({\lambda \over 1 +z_s},\Delta t_1-\Delta t_2\right).\nonumber \end{eqnarray} \tag{ 4 }$$

We have referred the equations for the magnification of both images to an arbitrary time, t (note that microlensing-induced variability between a pair of images is uncorrelated).

The first term of Equation (4) is the relative microlensing magnification between images 1 and 2. The significance of the second term, Δm^con₀, which represents the source variability, can be estimated by comparing the intrinsic quasar variability on timescales typical of the time delay between images in gravitational lens systems with the expected distribution of microlensing magnifications. As we shall discuss in Section 4, the intrinsic source variability is not significant for our computations.

In summary, with the proposed method similar information as in the Milky Way MACHO experiments is obtained but with a single-epoch measurement. The objective of this study is to apply this method to published data of quasar microlensing. In Section 2, we collect the data from the literature and fit models to the systems of multiply imaged quasars to derive suitable values of the projected matter density at the image locations. Using these values, probabilistic models for microlensing magnifications are derived in Section 3.1 for a range of fractions of mass in compact objects. Sections 3.2 and 3.3 are devoted to estimate this fraction. Finally, in Section 4 we present and discuss the main conclusions.

We collected the data, Δm (see Equation (4)), examining all the optical spectroscopy⁵ found in the literature (see Table 1). In most cases, the microlensing magnification or the scaling of the emission line ratio with respect to the continuum ratio are directly provided by the authors or can be estimated from a figure. For SDSS 0806+2006, FBQ 0951+2635, SDSS J1001+5027, QSO 1017 − 207, SDSS J1206+4332, HE 1413+117, and SBS 1520+530 we used the electronically available or digitized spectra of the images to estimate the microlensing magnification following the steps described in Mediavilla et al. (2005). In Table 1, we include (when available) the flux ratios for each line and its corresponding continuum. Specific details of the procedure followed to obtain the data are also given.

For some of the image pairs (∼30% of the sample) there are mid-IR flux ratios available. Except for one system, SDSS J1004+4112 (where image C is probably affected by extinction, Gómez-Álvarez et al. 2006), they are in very good agreement with the emission-line flux ratios (see Table 2). The average difference between mid-IR and emission line flux ratios is only 0.11 mag (0.07 mag if SDSS J1004+4112 is removed). In fact, the agreement is unexpectedly good taking into account the possible influence of extinction and source variability. In any case, this comparison supports the consistency of the basic hypothesis (that the emission line fluxes are not affected by microlensing) and the reliability of the data.

Figure 1 shows the frequency of observed microlensing magnifications, f_obs(Δm). This histogram exhibits two significant characteristics: the relatively high number of events with low or no microlensing magnification and the concentration (∼80%) of the microlensing events below |Δm| = 0.6. Any model attempting to describe microlensing magnification should account for these features. At a lower level of significance, the presence of two events of high magnification, Δm ∼ 1.5, should also be noted. The data presented in Table 1 come from many different bibliographic sources with the subsequent lack of information about measurement procedures and estimate of uncertainties. However, even with this limitation, the low frequency of high magnification microlensing events in the optical seems to be a reliable observational result.

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For each of the gravitational lens systems in Table 3 we have used a singular isothermal sphere plus external shear model (SIS+γ_e) to estimate values of the total projected matter density κ and the shear γ at the locations of the images (see Table 3). The models have been computed with the "lensmodel" code by Keeton (2001) to fit the positions of the images (CASTLES, http://www.cfa.harvard.edu/castles/). For double systems we have used the emission-line flux ratios between images as an additional constraint.

In Table 4, we show $n_{\kappa _1,\kappa _2}$ (κ₁ < κ₂), the frequency distribution of image pairs that occur at combined projected matter densities (κ₁, κ₂). The distribution peaks at bin (κ₁ = 0.45, κ₂ = 0.55). In many of the image pairs in Table 3 the images are roughly located at similar distances from the lens galaxy center, r₁ ∼ r₂. In an SIS model the convergence for each of the lensed images is given by κ₁ = 1/2(1 + x) and κ₂ = 1/2(1 − x), where x is the position of the source in units of the Einstein radius. The image configuration r₁ ∼ r₂ is obtained when x ≳ 0; therefore, the expected values for the convergence are κ₁ ≲ 0.5 and κ₂ ≳ 0.5. This is in agreement with Table 4 and in fact this simple reasoning could have been used to estimate, from a statistical point of view, the peak of the distribution of convergence values, $n_{\kappa _1,\kappa _2}$, in our sample.

Observational uncertainties in the flux ratios, differential extinction in the lens galaxy and more complicated mass distributions for modeling the lens galaxy could have an important impact on the estimates of κ and γ. Therefore, the values for κ and γ in Table 3 computed for an SIS+γ_e model should be taken individually only as compatible values with high uncertainties. However, we will assume that the distribution of values in its entirety can be considered as statistically representative for the sample of observed image pairs. Fortunately, the uncertainty in the macro-lens models does not play a crucial role in the conclusions of our study.

To analyze the microlensing magnification data, we need to consider that each Δm measurement results from the flux ratio of two images seen through different locations at the lens galaxy. The microlensing magnification probability of a given image, $f_{{\kappa _*}_1,\kappa _{1},\gamma _{1}}(m_1)$, depends on the projected matter density in compact objects, κ_*1, the total projected mass density, κ₁, and the shear, γ₁. Thus, the probability distribution of the difference in microlensing magnification of a pair of images, Δm = m₁ − m₂, is given by the integral

$$\begin{eqnarray} f_{{\kappa _*}_1,{\kappa _*}_2,\kappa _{1},\kappa _{2},\gamma _{1},\gamma _{2}}(\Delta m)&=&\int {f_{{\kappa _*}_1,\kappa _{1},\gamma _{1}}(m_1)f_{{\kappa _*}_2,\kappa _{2},\gamma _{2}}}\nonumber\\ &&{\times \, (m_1-\Delta m)d m_1}. \end{eqnarray} \tag{ 5 }$$

To simplify the analysis we will suppose that the fraction of matter in compact objects, α = κ_*/κ, is the same everywhere. The probability distribution of the difference in microlensing magnification of a pair of images can then be written as

$$\begin{eqnarray} f_{\alpha \kappa _{1},\alpha \kappa _{2},\kappa _{1},\kappa _{2},\gamma _{1},\gamma _{2}}(\Delta m)&=&\int {f_{\alpha \kappa _{1},\kappa _{1},\gamma _{1}}(m_1)f_{\alpha \kappa _{2},\kappa _{2},\gamma _{2}}}\nonumber\\ &&{\times \, (m_1-\Delta m)dm_1}. \end{eqnarray} \tag{ 6 }$$

From this expression we can evaluate the probability of obtaining a microlensing measurement Δm_i from a pair of images, $f^i_{\alpha \kappa ^i_{1},\alpha \kappa ^i_{2},\kappa ^i_{1},\kappa ^i_{2},\gamma ^i_{1},\gamma ^i_{2}}(\Delta m_i)$. Then, to estimate α using all the available information we maximize the likelihood function corresponding to the N measurements collected in Table 1,

$$\begin{equation} \log L(\alpha)=\sum _{i=1}^N{\log f^i_{\alpha \kappa ^i_{1},\alpha \kappa ^i_{2},\kappa ^i_{1},\kappa ^i_{2},\gamma ^i_{1},\gamma ^i_{2}}(\Delta m_i)}. \end{equation} \tag{ 7 }$$

3.1. Probability Distributions of the Difference in Microlensing Magnifications for Image Pairs, $f_{\alpha \kappa _{1},\alpha \kappa _{2},\kappa _{1},\kappa _{2},\gamma _{1},\gamma _{2}}$

We first compute the microlensing magnification probability distributions for one image, f_ακ,κ,γ(m). The first step is to simulate microlensing magnification maps for the different values of κ and γ in Table 3. We consider several values for the fraction of mass in compact objects⁶: α = 1, 0.5, 0.3, 0.25, 0.2, 0.15, 0.10, 0.05, 0.03, and 0.01. The histogram of each magnification map then provides a frequency distribution model of microlensing magnifications.

We obtain square maps 24 Einstein radii on a side with a spatial resolution of 0.012 Einstein radii per pixel. To compute the magnification maps we use the inverse polygon mapping method described in Mediavilla et al. (2006). An example of the maps is shown in Figure 2. The microlensing magnification at a given pixel is then obtained as the ratio of the magnification in the pixel to the average magnification. Histograms of these normalized maps give the relative frequency of microlensing magnifications for a pixel-size source (see some examples in Figure 3). These distributions are in agreement with the results obtained with a different method by Lewis & Irwin (1995).

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To model the unresolved quasar source we consider a Gaussian with r_s = 2.6 × 10¹⁵ cm (1 ld; Shalyapin et al. 2002; Kochanek 2004). The convolution of this Gaussian with the "pixel" maps gives the magnification maps for the quasar. For a system with redshifts z_l ∼ 0.5 and z_s ∼ 2 for the lens and the source respectively, the Einstein radius for a compact object of mass M is $\eta _0\sim 5.2 \times 10^{16}\sqrt{M/M_\odot }\;{\rm cm}$. Thus, for M = 1 M_☉, η₀ ∼ 5.2 × 10¹⁶ cm, and the size of a pixel is 6.2 × 10¹⁴ cm.

Finally, the histograms of the convolved maps give the frequency distributions of microlensing magnifications, f_ακ,κ,γ(m), which show differences with respect to the results obtained for a pixel-size source at the high magnification wing (the same effect that can be observed in Lewis & Irwin 1995). From the cross-correlation of pairs of these individual probability functions $f_{\alpha \kappa _1,\kappa _1,\gamma _1}(m_1)$, and $f_{\alpha \kappa _2,\kappa _2,\gamma _2}(m_2)$ (see Equation (6)) we obtain the probability function of the difference in microlensing magnification between two images, $f_{\alpha \kappa _{1},\alpha \kappa _{2},\kappa _{1},\kappa _{2},\gamma _{1},\gamma _{2}}(\Delta m=m_1-m_2)$. In Figures 4–6 the $f_{\alpha \kappa _{1},\alpha \kappa _{2},\kappa _{1},\kappa _{2},\gamma _{1},\gamma _{2}}(\Delta m)$ distributions corresponding to the 29 image pairs of Table 3 are plotted.

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3.2. Maximum Likelihood Estimate of the Fraction of Mass in Compact Objects, α: Confidence Intervals

In Figure 7, we present log L(α) (see Equation (7)). Using the log L(α ± nσ_α) ∼ log L_max − n²/2 criterion we derive α(log L_max) = 0.10^+0.04_−0.03 (90% confidence interval).

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The maximum likelihood method can be affected by errors in the microlensing measurements, $\sigma _{\Delta m_i}$. From Equation (7) we obtain

$$\begin{eqnarray} &&\hspace{-1.5pc}\Delta \log L(\alpha)\nonumber\\ &&\hspace{-1.5pc}\quad=\sum _{i=1}^N{{1 \over f^i_{\alpha \kappa ^i_{1},\alpha \kappa ^i_{2},\kappa ^i_{1},\kappa ^i_{2},\gamma ^i_{1},\gamma ^i_{2}}(\Delta m_i)} {\partial f^i_{\alpha \kappa ^i_{1},\alpha \kappa ^i_{2},\kappa ^i_{1},\kappa ^i_{2},\gamma ^i_{1},\gamma ^i_{2}}(\Delta m_i)\over \partial \Delta m_i}\sigma _{\Delta m_i}}.\nonumber\\ \end{eqnarray} \tag{ 8 }$$

According to this last expression, microlensing measurement errors do not significantly affect the likelihood of flat probability distributions (typical of large values of α). On the contrary, the likelihood functions corresponding to low values of α (associated with sharply peaked probability distributions) can be strongly modified by the microlensing measurement errors. Note, moreover, that these changes tend to penalize the low α hypothesis.

To show the impact of $\sigma _{\Delta m_i}$ on the maximum likelihood estimate of α, in Figure 7 we also present log L(α) (see Equation (7)) with error bars, ±Δlog L(α), estimated considering that each Δm_i is a normally distributed variable with $\sigma _{\Delta m_i}=0.20$ (a realistic estimate). Using the log L(α ± nσ_α) ∼ log L_max − n²/2 criterion and taking into account the error bars of log L(α), we derive α(log L_max) = 0.05^+0.09_−0.03 (90% confidence interval).

3.3. Influence of the Continuum Source Size, Influence of the Microlenses Mass

Increasing the size parameter of the Gaussian representing the continuum source, r_s, affects the previous results by smoothing the magnification patterns and, consequently, the probability distributions. To study the dependence of the estimate of α on the source size we have computed probability and likelihood functions for several values of this parameter, r_s = 0.62 × 10¹⁵, 2.6 × 10¹⁵, 8 × 10¹⁵, and 26 × 10¹⁵ cm. To correct r_s from projection effects we have taken into account that the intrinsic and projected source areas are related by a cos i factor; that is $r_s\sim \sqrt{\cos i}\, r_{s_0}$. Assuming that the (disk) sources are randomly oriented in space (the probability of finding a disk with inclination, i, proportional to sin i) and averaging on the inclination, we obtain $r_{s_0}\sim 1.5 r_s$. In Figure 8, we present the likelihood functions corresponding to sources of several deprojected size parameters, $r_{s_0}$. In Figure 9, we plot the maximum likelihood estimate of α versus⁷$r_{s_0}$. Error bars correspond to 90% confidence intervals. According to this figure, low values of α are expected for continuum source sizes, $r_{s_0}$, of the order of 10¹⁶ cm or less. Observing microlensing variability for nine gravitationally lensed quasars Morgan et al. (2007) measure the accretion disk size. The average value of the nine half-light radius determinations is 〈r_1/2〉 = 6 × 10¹⁵ cm. For this value we found (see Figure 9) α = 0.05^+0.09_−0.03. Morgan et al. (2007) report a scaling between the accretion disk size and the black hole mass. In the range of black hole masses considered by Morgan et al. (2007) the maximum is M_BH = 2.37 × 10⁹ M_☉, which, using the scaling derived by these authors, corresponds to r_1/2 = 2.4 × 10¹⁶ cm. For this size we obtain (see Figure 9) α ∼ 0.10. Values M_BH ⩾ 10¹⁰ M_☉ (r_1/2 ⩾ 3.4 × 10¹⁶ cm) should be considered to obtain α ⩾ 0.20. On the other hand, Pooley et al. (2007) comparing X-ray and optical microlensing in a sample of 10 lensed quasars inferred r_1/2 ∼ 1.3 × 10¹⁶ cm. For this size we obtain (see Figure 9) α = 0.10^+0.05_−0.06. Thus, according to these recent size estimates based on the observations of two relatively large samples of gravitational lenses, high values of α are possible only if the continuum source size is substantially larger than expected.

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Owing to the scaling of the Einstein radius with mass, $\eta _0\propto \sqrt{M}$, a change in the mass of microlenses can be alternatively seen as a change in the spatial scaling of the magnification pattern that leaves invariant the projected mass density, κ. Thus, multiplying the mass of the microlenses by a factor C (and leaving unaltered the continuum size) is equivalent to multiplying the size of the continuum source by a factor $1/\sqrt{C}$ (leaving unaltered the masses of microlenses). Then the computed models corresponding to sources of sizes r_s = 0.62 × 10¹⁵, 2.6 × 10¹⁵, and 8 × 10¹⁵ cm (with 1 M_☉ microlenses) are equivalent to models corresponding to microlens masses of 17, 1, and 0.1 M_☉ (with r_s = 2.6 × 10¹⁵ cm). This result implies that the probability models do not differ significantly if we change the mass of microlenses between 17 and 0.1 M_☉. Thus, microlensing statistics are insensitive to changes of mass in the expected range of stellar masses.

In the previous sections, we have extended to the extragalactic domain the local (Milky Way, LMC, and M31) use of microlensing to probe the properties of the galaxy halos. Although our primary aim was to explore the practical application of the proposed method, we found that the data available in the literature can be consistently interpreted only under the hypothesis of a very low mass fraction in microlenses; at 90% confidence: α(log L_max) = 0.05^+0.09_−0.03 (maximum likelihood estimate) for a quasar continuum source of intrinsic size r_s = 2.6 × 10¹⁵ cm. This result arises directly from the shape of the histogram of microlensing magnifications, with a maximum of events close to no magnification and stands for a wide variety of microlensing models statistically representative of the considered image pairs. There is a dependence of the estimate of α on the source size but high values of the mass fraction (α>0.2) are possible only for unexpectedly large source sizes (r_s > 4 × 10¹⁶ cm). The low mass fraction is in good agreement with the results of EROS (Tisserand et al. 2007) for the Milky Way, with the estimate of OGLE for the LMC (Wyrzykowski et al. 2009), and with the limit established by MEGA (de Jong et al. 2006) for M31. The agreement is also good with the few microlensing-based estimates available for individual objects. In RXJ 1131 − 1231, Dai et al. (2009) found α ∼ 0.1. In PG 1115+080, Morgan et al. (2008) obtained values in the range α = 0.08–0.15. For the same system, Pooley et al. (2009) found α ∼ 0.1 for a source of size r_s = 1.3 × 10¹⁶ cm.

On the other hand, our estimate of the fraction of mass in microlenses, α(log L_max) = 0.05^+0.09_−0.03, approximates the expectations for the fraction of visible matter. Jiang & Kochanek (2007), for instance, comparing the mass inside the Einstein ring in 22 gravitational lens galaxies with the mass needed to produce the observed velocity dispersion, inferred average stellar mass fractions of 0.026 ± 0.006 (neglecting adiabatic compression) and 0.056 ± 0.011 (including adiabatic compression). As discussed in Jiang & Kochanek (2007) these values are also in agreement with other estimates of the stellar mass fraction that relied on stellar population models: ∼0.08 (Lintott et al. 2006), 0.065^+0.010_−0.008 (Hoekstra et al. 2005), and 0.03^+0.02_−0.01 (Mandelbaum et al. 2006). Thus, we can conclude that microlensing is probably caused by stars in the lens galaxy, and that there is no statistical evidence for MACHOS in the halos of the 20 galaxies of the sample we considered.

How robust are these results? There are several sources of uncertainty to consider. First, we neglected in Equation (4) the term arising from source variability, Δm^con₀. From a group of 17 gravitational lenses with photometric monitoring available in the literature we estimate an average gradient of variability of 0.1 mag year⁻¹. Taking into account that the average delay between images is about three months (a conservative estimate; note that the group of lens systems used includes many doubles, some of them with very large time delays) we can expect an amplitude related to intrinsic source variability of Δm^con₀ ∼ 0.03, which, according to the histogram of magnifications (Figure 1), is not significant. Moreover, if we assume that the probability of Δm^con₀ is normally distributed, the global effect of source variability is to broaden the histogram of microlensing magnifications, diminishing the peak and enhancing the wing. In other words, source variability leads to an overestimate of α. Thus, the mass fraction should be even lower if significant source variability were hidden in the data. In the same way, other sources of error in the data, such as the difficulty in separating line and continuum or in removing from the narrow emission lines the high ionization broad emission lines that could be partially affected by microlensing, probably tend to induce additional magnitude differences, Δm, between the images and, hence, to an overestimate of α. On the contrary, cross-contamination between the spectra of a pair of images masks the impact of microlensing and may affect our results. Although most of the bibliographic sources of microlensing measurements analyze this problem concluding that the spatial resolution was sufficient to extract the spectra without contamination, it is clear that high signal-to-noise ratio (S/N) data obtained in subarcsecond seeing conditions will help to control this important issue.

Another point to address is the treatment of some of the quads, where only a subset of the images are used. Are we systematically excluding faint images that might be highly demagnified by microlensing? Let us examine the four incomplete quads in our sample. The fold lens SDSS J1004+4112 has two close images A and B. A is probably a saddle-point image and shows the most anomalous flux (Ota et al. 2006). In contrast, the optical/X-ray flux ratios of C and D are almost the same. Thus, there is no reason to suppose that the image without a useful spectrum (D) has higher microlensing probability than the others. PG 1115+080 is another fold quad. A₁ and A₂ are the two images closest to the critical curve and have a (moderately) anomalous flux ratio and optical variability (Pooley et al. 2007). The two images without available spectra (C and D) show only a small optical variability and are not particularly prone to microlensing. In RXS J1131 − 1231, the most anomalous flux ratio is B/C and A is a saddle-point image (Sluse et al. 2006). Image D (the one with no available spectrum) also has an anomalous flux but is not more susceptible to microlensing than the other images. Thus, in three of the four incomplete quads there is no reason to suppose that we are biasing the sample toward image pairs with lower microlensing probability. The case of SDSS 0924+0219 is more problematic. There are two sets of data for this object, one by Eigenbrod et al. (2006) based on observations of the low ionization lines [Mg ii] and [C iii], which, after two epochs of observation, reveals no difference between the line and continuum flux ratios of components A and B. The other set of data (Keeton et al. 2006) is based on Lyα observations (a high ionization emission line supposed to come from a smaller region than the low ionization emission lines) and microlensing is detected not only in the continuum but also in the emission lines. This implies that the baseline for no microlensing magnification cannot be defined and, consequently, we could not consider Keeton et al. (2006) results. Anyway, we have repeated (as a test) the entire maximum likelihood estimate procedure to derive α but now using for SDSS 0924+0219 the microlensing measurements by Keeton et al. (2006). The results are almost identical: α = 0.05^+0.10_−0.03.

The size of the sample also limits the statistical interpretation. An improvement in the S/N of the histogram of microlensing magnifications is very important to ascertain the statistical significance of the low frequency of events at large magnification (only two events of high magnification are detected), which can impose severe constraints on the microlensing models. Another reason to increase the size of the sample is the possibility to define subsamples at different galactocentric distances where different ratios of visible to dark matter are expected. In the same way, it would be possible to define subsamples according to the type of lens galaxy or other interesting properties of lens systems.

In any case, the impact of the main result of our study—absence of MACHOS in the 10–0.1 M_☉ mass range in the halos of lens galaxies—and its future prospects, points to the need to improve the statistical analysis in two ways: increasing the number, quality, and homogeneity of the microlensing magnification measurements from new observations, and reducing the uncertainties in the macro-lens models.

We thank the anonymous referee for valuable comments and suggestions. We are grateful to A. Eigenbrod, P. Green, M. Oguri, L. Wisotzki, and O. Wucknitz for kindly providing spectra. This work was supported by the European Community's Sixth Framework Marie Curie RTN (MRTN-CT-505183 "ANGLES") and by the Ministerio of Educación y Ciencia of Spain with the grants AYA2004-08243-C03-01 and AYA2004-08243-C03-03. V.M. acknowledges support by FONDECYT grant 1071008. J.A.M. is also supported by the Generalitat Valenciana with the grant PROMETEO/2009/64.