Theoretical expectation

For a QSO number count of the form $N(<m)\propto10^{\alpha m}$ , gravitational lensing causes a cross-correlation between galaxies and QSOs of the form $\omega_{\rm qg}(\theta)=b_{\rm g}(2.5\alpha-1)\omega_{\mu\delta}(\theta)$ in the weak lensing regime. Note, we assume that the intrinsic number count slope is not significantly modified by the lensing, this is reasonable given the flat number count slope of the faint QSOs considered here (Hamana, Martel & Futamase 2000). $b_{\rm g}$ is the linear bias of the foreground galaxies and $\omega_{\mu\delta}(\theta)$ is the cross correlation function between the magnification, $\mu$ , and the density contrast, $\delta$ . Bartelmann (1995) has shown (see also Bartelmann & Schneider 1999) that $\omega_{\mu\delta}(\theta)$ is an integral over the mass power spectrum, P(k), and the radial distributions of the QSOs and galaxies.

The observed galaxy auto-correlation functions is $\omega_{gg}=b_g^2\omega_{\delta\delta}$ . The mass correlation function, $\omega_{\delta\delta}$ , is an integral over P(k) and the radial distribution of galaxies (Limber 1953). When we take the ratio $\omega_{\mu\delta}/\omega_{\delta\delta}$ , to first order the integrals over the P(k) cancel such that the ratio is constant (to $\sim1-2$ per cent) as a function of $\theta$ on scales of interest. This makes it easy to compare QSO-galaxy cross-correlations to galaxy-galaxy auto correlations via

$\begin{displaymath} \frac{\omega_{\rm qg}(\theta)}{\omega_{\rm gg}(\theta)}=\fra... ...ac{\omega_{\mu\delta}(\theta)}{\omega_{\delta\delta}(\theta)}. \end{displaymath}$

(1)

**Figure 2:** Angular cross-correlation function between QSOs with $z\leq 1$ and galaxies limited to $B_{\rm ccd}<23$ (a) and 26 (b). The data are are plotted after inclusion of the integral constraint. The dotted line indicates the amplitude of the integral constraint, which is too small to be visible in b. The dashed line in a is the best fit -0.8 power law.
$\begin{figure} \centering\centerline{\psfig{file=fig2.ps,width=16.0cm}}\end{figure}$

We assume a $\Gamma_{\rm eff}\simeq\Omega h=0.25$ CDM power spectrum, allowing for non-linear effects using the empirical fits of Peacock & Dodds (1996). We also assume that $b_{\rm g}$ denotes the average linear bias of the population considered. We use the galaxy redshift distributions shown in Fig. 1, which are analytic models of the form $N(z)\propto z^2\exp[-(z/z_{\rm c})^{\beta}]$ (e.g. Baugh & Efstathiou 1993), with parameters $z_{\rm c}$ and $\beta$ chosen to match the redshift distributions found by Glazebrook et al. (1995) at B<23 ( $\beta\simeq1.5$ and $z_{\rm c}\simeq0.25$ ), and Fernández-Soto, Lanzetta & Yahil (1999) at B<26 ( $\beta\simeq1.15$ and $z_{\rm c}\simeq0.5$ ). We note that the exact form of the redshift distribution make little difference to the expected lensing amplitude. The above redshift distribution for B<23 gives $\omega_{\mu\delta}/\omega_{\delta\delta}\simeq0.21$ and 0.09 for the EdS and $\Lambda$ cosmologies respectively. We use the Infante & Pritchet (1995) measurement of the auto-correlation function of B<23 galaxies, $\omega_{\rm gg}(\theta)=(0.045\pm0.004)\theta^{-0.8}$ with $\theta$ in arcmins, to make our comparisons. We fit a -0.8 power law to the observed $\omega_{\rm qg}$ to determine the ratio $\omega_{\rm qg}/\omega_{\rm gg}$ and thus derive $(2.5\alpha-1)/b_{\rm g}$ via Eq. 1.

An alternative explanation for a cross-correlation signal is that dust associated with the foreground galaxies causes extinction in the QSOs, so that less are found nearby the galaxies. The extinction in the B-band is $A_{\rm B}$ , with $A_{\rm B}=x(z)E(B-V)$ , where E(B-V) is the measured reddening, and $x(z)=A_\lambda/E(B-V)$ is a function of the redshift of the absorbing material. At z=0 x(0)=4.0 for absorption in the B-band. At higher redshift, the observed B-band is moved into the UV, so that for a given column of dust, the extinction will be greater. When integrating over the n(z) distribution for B<23 galaxies the mean value of $A_\lambda/E(B-V)$ for the observed B-band is 5.57. If $\alpha A_{\rm B}\ll1$ then the cross-correlation due to inter-galactic dust is $\omega_{\rm qg}(\theta)\simeq-\alpha A_{\rm B}\ln(10)$ , with $A_{\rm B}$ a function of $\theta$ . The amount of dust required depends on the steepness of the QSO number counts slope.