All Figures

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{theoretical_kk.ps} \end{figure}$ Figure 1: The upper panel shows the amplitude of the two first terms of the Taylor expansion of the magnification autocorrelation, namely $\langle\kappa(\vec\phi)\kappa(\vec\phi+\vec\theta)\rangle$ (dashed line) and $4\langle\kappa(\vec\phi)\kappa^2(\vec\phi+\vec\theta)\rangle$ (dotted line), using a source redshift of unity. The sum of these two terms is shown as the solid line. The lower panel details the relative contribution $R_{\mu \mu }$ of the second-order term for different source redshifts. The figure shows that the lowest-order approximation $\mu \approx 1+2\kappa$ misses a substantial part of the amplitude of the magnification autocorrelation function. Given the accuracy of the bispectrum fitting formula, $R_{\mu \mu }$ is accurate to $\sim$ 2%.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{pdf_mu3.ps} \end{figure}$ Figure 2: Probability distribution of the magnification for our small- and large-scale smoothing simulations, assuming sources at redshift unity. The power law tail behaviour ( $\propto$ $\mu ^{-2}$ ) found in the small-scale smoothing indicates the existence of caustics, while for large-scale smoothing, no prominent tail is shown which suggests caustics do not play a noticeable role.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{smoothing.ps} \end{figure}$ Figure 3: Smoothing angle of the simulation as a function of redshift for the two ray-tracing schemes. In order to show the relevant quantities leading to the effective smoothing angle, we overplot the weighting function $W(z)=D_+^4(z)~p_\kappa^3(z)$ (see Eqs. (8) and (21)).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{comparison_numerical.ps} \end{figure}$ Figure 4: Comparison between theoretical predictions and measurements from numerical simulations assuming sources at redshift unity. The upper and lower curves show $\langle\kappa\kappa\rangle$ and $\langle\kappa\kappa^2\rangle$ , respectively. The points are measurements from the large-scale smoothing simulations, with the error bars showing the variance among 36 different realisations. The dotted lines show the analytical computations taking into account the smoothing scale of the simulation. The solid lines additionally include a cut in the power spectrum for cancelling the wavelengths not covered by the simulation. The dashed line presents the same statistics without any smoothing. Obviously, the smoothing effects are crucial for the $\langle\kappa\kappa^2\rangle$ cross-correlation.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{comparison_mu_longshoot.ps} ... ...mm} \includegraphics[width=8cm,clip]{comparison_mu_raytrix_mask.ps} \end{figure}$ Figure 5: The upper panel shows a comparison between the magnification autocorrelation measured from the simulation compared to the analytical estimation for a source redshift of unity. The circles show averaged measurements from 36 realisations of the simulation and the corresponding 1- $\sigma$ error bars. The dashed line shows the analytical estimation using the approximation $\mu \approx 1+2\kappa$ . The solid line shows the improvement given by the second-order term of the Taylor expansion of $\mu$ . The lower panel shows the relative contribution $R_{\mu \mu }$ , both measured from the simulation and estimated using expansion terms of $\mu$ up to second order. In the right panel, each angular point shows three different measurements, taking into account the complete magnification maps (solid-line error bars), or maps where pixels were masked where the magnification value exceeds 8 or 4 (dashed-line error bars, from left to right).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{w_qg_corrected.ps} \end{figure}$ Figure 6: The upper panel shows the amplitude of the normalised quasar-galaxy correlation $w_{\rm QG}/2(\alpha -1)$ as a function of angular separation. We show the first two terms of the Taylor expansion of this correlation, namely $\langle \delta(\vec\phi)\kappa(\vec\phi+\vec\theta)\rangle$ (dashed line) and $2\langle\delta(\vec\phi)\kappa^2(\vec\phi+\vec\theta)\rangle$ (dotted line), using a source redshift of unity. The sum of these two terms is shown as the solid line. The lower panel details the relative contribution $R_{\delta \mu }$ and of the second-order term for different source redshifts, namely z=1, 1.5, 2 and 3 from bottom to top.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{comparison_mu_delta.ps} \end{figure}$ Figure 7: Comparison of $R_{\delta \mu }$ given by the theoretical calculation and measured from the numerical simulation. The results are shown for the large- and small-scale smoothing simulations from bottom to top. They show that a second-order description of the $\langle\mu\delta\rangle$ cross-correlation gives good results when the smoothing is large, but misses some part of the total amplitude in the case of our small-scale smoothing ray-tracing.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{pdf_mu3.ps} \end{figure}$	Figure 2: Probability distribution of the magnification for our small- and large-scale smoothing simulations, assuming sources at redshift unity. The power law tail behaviour ( $\propto$ $\mu ^{-2}$ ) found in the small-scale smoothing indicates the existence of caustics, while for large-scale smoothing, no prominent tail is shown which suggests caustics do not play a noticeable role.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{smoothing.ps} \end{figure}$	Figure 3: Smoothing angle of the simulation as a function of redshift for the two ray-tracing schemes. In order to show the relevant quantities leading to the effective smoothing angle, we overplot the weighting function $W(z)=D_+^4(z)~p_\kappa^3(z)$ (see Eqs. (8) and (21)).
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