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the gravitational deflection of photons can be used as a probe of the matter distribution along the line of sight to the sources . the latter may be at the last scattering surface ( @xmath1 ) , in the case of the cosmic microwave background ( seljak 1996 ; martnez - gonzlez et al . 1997 ; bernardeau 1997 ) , or high@xmath2 objects as qsos or galaxies ( blanford et al . 1991 ; kaiser 1992 ; kaiser & squires 1993 ; bartelmann 1995 ; villumsen 1995b ; villumsen 1996 ; bernardeau , van waerbeke & mellier 1996 ; kaiser 1996 ; jain & seljak 1996 ) . information about the matter fluctuations can be obtained on different scales ranging from galaxy haloes to the large - scale structure of the universe . many of the theoretical studies on this subject have dealt with the polarization or ellipticity produced on background galaxies by the large - scale structure of the universe , and there are currently several ongoing observational projects trying to detect and quantify this effect . nevertheless , measuring shear amplitudes as the ones predicted by the above mentioned calculations is very difficult from a technical point of view ( although see mould et al . 1991 ; villumsen 1995a ) , and it is not totally clear if such a precision would be routinely achievable in the near future ( ringberg workshop 1997 ) .    however , there is another observable phenomenon produced by gravitational lensing of background sources by foreground mass distributions which may have been already detected : qso - galaxy associations due to the magnification bias effect @xcite . the surface density of a sample of flux - limited background sources behind a lens which magnifies them by a factor @xmath3 is changed in the form @xmath4 , where @xmath5 is the unperturbed background source density . if @xmath6 ( or @xmath7 ) , the change in the density can be characterized by the factor @xmath8 . thus , depending on the slope @xmath9 there may be an excess of background sources ( @xmath10 ) , a depletion ( @xmath11 ) , or the density may remain unchanged ( @xmath12 ) . if we properly choose the background source population , so that it has a slope @xmath9 considerably different from 1 , there would be a correlation ( or anticorrelation ) between the position of the matter overdensities acting as lenses and the background sources . now , these matter perturbations will be traced , up to a bias factor , by galaxies and thus , there will be a correlation between these foreground galaxies ( or any other tracers of dark matter ) and the background sources . there are several reported associations between foreground galaxies and high redshift , background agns ( see schneider , ehlers & falco ( 1992 ) ; narayan & bartelmann ( 1996 ) or wu ( 1996 ) for reviews ) , but only a few of these studies extend to relatively large scales . bartelmann & schneider ( 1994 ) found a strong association between galaxies selected from the iras faint source catalogue and high@xmath2 agn from the 1jy catalogue . in bentez & martnez- gonzlez ( 1995 ) it was found that red apm galaxies tracing large scale structures were correlated with 1jy qsos . another sample of radio loud qsos , extracted from the pks catalogue has also been found to be correlated with cosmos galaxies @xcite , with a correlation scale of several arcmin . other studies considering the correlation between galaxy clusters and high - z qsos ( seitz & schneider 1995 , wu & han 1996 ) have also found positive results .    in this paper , we shall study the effects of weak gravitational lensing by foreground matter fluctuations on a population of background sources at high@xmath2 . we consider different values of @xmath0 and model the fluctuations assuming cdm with a power spectrum whose evolution in time follows a standard ansatz ( hamilton et al . 1991 , peacock & dodds 1996 , linear and non - linear contributions are considered ) . we assume that these matter perturbations are traced , up to a global bias parameter @xmath13 by galaxies . more specifically , we shall explore the behavior of @xmath14 , i.e. the large - scale correlation between the ellipticity of background galaxies and the position of foreground ones , which apparently has not been considered in the literature . we shall also consider in detail other correlations ( in particular their dependence on @xmath0 ) such as @xmath15 i.e. magnification - foreground galaxies and magnification - magnification . @xmath16 can be indirectly estimated through the galaxy - galaxy correlation function ( villumsen 1995b ) . however , measuring @xmath17 offers several advantages over @xmath16 from the observational point of view . in the first place , @xmath17 has an amplitude several times higher than @xmath16 . besides , if the foreground and background galaxy populations are properly selected so that there is no redshift overlap between them ( e.g high@xmath2 qsos and bright galaxies ) , one does not have to bother about intrinsic correlations : any measured effect should be caused by gravitational lensing . section 2 develops the formalism dealing with weak gravitational lensing for a flat and open cosmological model , the concepts of magnification and polarization ( or ellipticity ) and the different correlations . in section 3 appear the main theoretical results as well as comments on different observational perspectives . finally , in section 4 we give the conclusions of the paper . we will consider the propagation of photons from a source at redshift @xmath18 to the observer ( @xmath19 ) , the universe being a perturbed friedmann model with vanishing pressure . for scalar perturbations , the metric in the conformal newtonian gauge is given in terms of the scale factor @xmath20 and a single potential @xmath21 , that satisfies the poisson equation , as follows ( martnez - gonzlez et al . 1997 ) @xmath22,\ ] ] @xmath23 we take units such that @xmath24 and @xmath25 denote the flat , open and closed friedmann background universe .    assuming a perturbation scheme ( `` weak lensing '' ) , the null geodesic equation for the previous metric can be integrated in the form @xmath26 , where @xmath27 is the direction of observation and @xmath28 is the distance to the photon in the background metric , i.e.      the perturbation @xmath30 can be decomposed in a term parallel to the direction of observation @xmath27 and a term , @xmath31 , ortoghonal to such a direction . the last term is given by    @xmath32    where @xmath33 is a window function    @xmath34    for photons that are propagated from a source at redshift @xmath18 ( distance @xmath35 ) to the observer(@xmath36 or @xmath37 ) , the lensing vector @xmath38 is defined in the usual way : @xmath39 thus we find @xmath40 once we have obtained the expression for the trajectory of the photon in the conformal newtonian gauge , it is easy to calculate everything in the conformal synchronous - comoving gauge ( martnez - gonzlez et al . the lensing vector in such a gauge ( that is the appropriate one from the point of view of observations ) is given by the expression ( 5 ) plus some additional terms that can be interpreted as doppler contributions at the source and observer and an acceleration term at the observer . the last two terms can be estimated from the doppler velocity respect to the cosmic microwave background and from our local infall towards the virgo cluster ( or great attractor ) . these extra contributions are very small , so the lensing vector @xmath38 in the synchronous - comoving gauge is approximately given by @xmath38 , as defined by equations ( 3 , 5 ) . let us assume a population of background sources ( e.g. quasars or galaxies ) , placed at different distances @xmath35 with a distribution @xmath41 ( @xmath42 ) . then , we can define the integrated lensing vector @xmath43 and taking into account equations ( 3 - 5 ) we obtain    @xmath44    @xmath45 @xmath46    if we take the derivative along the plane orthogonal to the direction of observation @xmath27 , we get the convergence tensor @xmath47 which can be decomposed in the form    @xmath48    for weak lensing , the convergence scalar @xmath49 is related to the magnification by @xmath50 . moreover , the polarization components in the plane orthogonal to @xmath27 can be defined in the standard way @xmath51 and the complex polarization is @xmath52 . so , we can define the scalar polarization @xmath53 . let us consider a second population of foreground sources ( e.g. galaxies ) placed at different distances @xmath28 with a distribution @xmath54 ( @xmath55 ) . if @xmath56 represents the density fluctuation ( that satisfies the poisson equation @xmath57 ) , we shall assume that there is a constant bias factor @xmath13 relating the number fluctuation , @xmath58 , and such an overdensity . then , we can define the integrated overdensity @xmath59 . taking into account equations ( 3 - 7 ) , we obtain    @xmath60    @xmath61    where @xmath62 . using the limber approximation ( see appendix ) , i. e. only a small region @xmath63 is contributing with @xmath64 , the previous equation can be approximated by    @xmath65\times\nonumber\\   & \times&{\int}_{\theta s}^{\infty}dr \frac{c_{\phi \delta}(\lambda ; r ) } { { ( r^2 - { \theta}^2 s^2)}^{1/2}},\end{aligned}\ ] ]    where now appears the correlation at a single time and @xmath66 is given by @xmath67^{-1}$ ] . introducing the power spectrum @xmath68 of the matter density fluctuations defined by    @xmath69    and the relation @xmath70 , obtained via the poisson equation for small scales , the last expression ( 11 ) is    @xmath71    finally , taking another @xmath72 derivative on the equation ( 9 ) , we get the following correlations    @xmath73    @xmath74    @xmath75    and @xmath76 is the bessel function of 1st kind . these are the basic formulas to be applied when two different populations are correlated . in particular , the background - foreground correlation function is given by bartelmann ( 1995 ) .    @xmath77    where @xmath78 is the slope of the background source number counts . if the background population is concentrated at a certain redshift @xmath79 corresponding to a distance @xmath80 , having a dirac delta distribution @xmath81 , then the window @xmath82 is given by @xmath83 @xmath84 if the foreground population has also a dirac delta distribution form @xmath85 , then the window @xmath82 in equation ( 16 ) is @xmath86 and @xmath87 } { \pi\lambda_b(1-\lambda_f)^2[1-(1-\omega)\lambda_f^2]}\times\nonumber\\ & \times&\int_0^\infty dk k p(\lambda_f , k)j_i(k\theta s_f)\end{aligned}\ ] ] where @xmath88^{-1}$ ] . let us consider a population of background sources ( e.g. galaxies ) placed at different distances @xmath28 with a distribution @xmath41 ( @xmath89 ) . we are now interested in the gravitational lensing properties induced by the population itself . the magnification - magnification @xmath90 , polarization - polarization @xmath91 and magnification - overdensity @xmath92 correlations are defined by    @xmath93    @xmath94    @xmath95    after an straightforward calculation , one obtains    @xmath96}^3}{{(1 - \lambda ) } ^4}\times\nonumber\\ & \times&\int_0 ^ 1dkk^{-3}p(\lambda , k)j_0(ks\theta ) , \end{aligned}\ ] ]    where the limber approximation has been assumed and @xmath82 is given by equation ( 7 ) . now , taking into account the definitions for the correlations , one gets ( see also kaiser 1992 , 1996 ; jain & seljak 1996 ) @xmath97}\times\nonumber\\ & \times&\int_0^\infty dkkp(\lambda , k)j_0(ks\theta ) , \end{aligned}\ ] ] @xmath98    the last expression in agreement with villumsen ( 1995b ) . for a population with a dirac delta distribution @xmath99 , the window @xmath100 is given by equation ( 18 ) . moreover , @xmath101 is given by equation ( 14 ) but now @xmath102 must be substituted by @xmath103 . trivially , for a strongly peaked @xmath18-distribution such a correlation is very small ( see the dependence on @xmath104 in equation ( 19 ) ) . it is interesting to note that the observed correlation function @xmath105 associated to a population is given by the following contributions ( villumsen 1995b )    @xmath106    @xmath107    where @xmath108 is the matter correlation function . so , gravitational lensing modifies the intrinsic correlation approximately by the magnification term . with the formalism presented in the previous section we have calculated the correlations @xmath17 , @xmath14 and @xmath16 . we assume a cdm model with a primordial harrison - zeldovich spectrum , a hubble parameter h@xmath109 ( h@xmath110h km s@xmath111 mpc@xmath111 ) and flat as well as open universe models . for the power spectrum we have used the fit given by equation ( g3 ) of bardeen et al . ( 1986 ) which is normalized to the cluster abundance : @xmath112 , following viana and liddle ( 1996 ) ( see also white , efstathiou and frenk 1993 ; eke , cole and frenk 1996 ; see kaiser 1997 for a discussion on alternative normalizations ) . for the nonlinear evolution of the power spectrum we use the recently improved fitting formula given by peacock and dodds ( 1996 ) . that formula is based on the hamilton et al . ( 1991 ) scaling procedure to describe the transition between linear and nonlinear regimes . it accounts for the correction introduced by jain , mo and white ( 1995 ) for spectra with @xmath113 and applies to flat as well as open universes .    for the redshift distributions of the background and foreground sources we consider a dirac delta distribution peaked at @xmath114 and @xmath115 respectively , @xmath116 and @xmath117 . these simple distributions are very useful since they reduce the calculations and the results differ only slightly when compared to other more realistic distributions ( see below ) . @xmath17 and @xmath14 are computed using equations ( 14,15,19 ) . @xmath118 is obtained from equation ( 25 ) where the function @xmath82 is given by equation ( 18 ) . the crosscorrelation @xmath119 for a population of background sources peaked at @xmath120 and another of foreground lenses at @xmath121 is given in figure ( 1 ) . the effect is maximum at zero lag and rapidly decreases for scales above a few arcmin . the amplitude is always above a few percent for scales @xmath122 and at these scales the linear contribution is negligible compared to the nonlinear one for all the @xmath0 values . notice that @xmath17 increases when @xmath0 decreases whereas considering only the linear contribution the situation is reversed .    from the observational point of view it is useful to calculate the average crosscorrelation ( or equivalently the mean relative excess of background sources around foreground lenses ) within a given radius @xmath123 , @xmath124 . the variation of @xmath125 with @xmath126 for different values of @xmath79 and a @xmath127 radius is shown in figures ( 2a , b , c ) . a maximum amplitude of a few percent is obtained at a @xmath126 in the range @xmath128 for all background populations and all cosmological models . ( at this point it is important to recall that to compare with observations of background - foreground object correlations @xmath129 , the bias factor @xmath13 of the foreground population and the slope @xmath9 of the background population enter in the calculation following equation 16 ) . it is interesting to notice that there are already available large galaxy samples , like the apm or cosmos catalogues , which peak at a redshift within that range ( for the apm catalogue @xmath130 for a magnitude limit @xmath131 , see efstathiou 1995 ) . moreover , the use of realistic redshift distributions to represent the foreground and background populations ( with a bell - like shape similar to that of the apm one ) , changes the results in only @xmath132 for the relevant angular scales compared to the dirac delta distribution used here . those catalogues , which except for a bias factor are assumed to follow the large scale matter distribution , are therefore very suitable to crosscorrelate with a background source population . this has already been done by bentez and martnez - gonzlez ( 1995 , 1997 ) for the 1 jansky and parkes samples of radio loud qsos as background populations , finding clear evidences of positive crosscorrelations . in figure 3 , we show @xmath119 for @xmath133 and for @xmath134 and @xmath120 , mean redshift values appropriate for the galaxy and radio qso samples considered above . the dependence of @xmath17 with @xmath0 is more relevant at small angular scales . in figure 4 , we also represent the average crosscorrelation @xmath125 as a function of @xmath0 . a rough comparison between its expected amplitude and the measured value for the cosmos - parkes samples , as given in figure 4 of bentez and martnez - gonzlez 1997 , shows agreement for realistic values of @xmath0 and the bias parameter . a detailed comparison of the theoretical calculations with the observational results will be given elsewhere . dolag and bartelmann ( 1997 ) have recently presented calculations of the qso - galaxy correlation function for such qso and galaxy populations produced by gravitational lensing due to the large scale structure , following a similar theoretical scheme . the crosscorrelation between the polarization of a background source population peaked at @xmath79 and the matter density fluctuations peaked at @xmath126 , @xmath135 , is given in figure ( 5 ) for @xmath120 and @xmath136 . the maximum value is in the angular range @xmath137 and this scale is smaller for low @xmath0 models . as in the case of @xmath17 , the linear contribution is negligible ; however , it peaks at a much larger angle of @xmath138 as a consequence of the much larger scales which contribute to the linear level . including the nonlinear evolution , a correlation of @xmath139 can be expected at angular scales @xmath122 for realistic models of structure formation . the use of realistic redshift distributions to represent the foreground and background populations ( with a bell - like shape typical of magnitude limited samples ) changes the results in only @xmath132 for the relevant angular scales compared to the dirac delta distribution used here .    in analogy to the previous subsection , it is useful to calculate the average crosscorrelation within a given radius @xmath123 , @xmath140 . the variation of @xmath141 with @xmath126 for several values of @xmath79 and @xmath0 is shown in figures ( 6a , b , c ) . maximum amplitudes of the order of @xmath142 are found for @xmath126 within a relatively wide range @xmath143 . the amplitude grows appreciably with @xmath79 being a factor of @xmath144 difference between @xmath145 and @xmath146 . considering the behavior of @xmath140 from these figures , suitable populations to detect the crosscorrelation would be a foreground sample with @xmath147 and a background one peaked at @xmath148 . in this subsection we concentrate on a single population of background sources peaked at a given @xmath79 and calculate @xmath149 . this is done in figure ( 7 ) for @xmath120 . the maximum effect is at zero lag and its amplitude is relatively small @xmath150 . the nonlinear contribution clearly dominates over the linear one but now the amplitude grows with @xmath0 , contrary to the crosscorrelations @xmath119 and @xmath135 . our results are in agreement with jain & seljak ( 1996 ) , and with kaiser ( 1997 ) for one degree scales ( see also those papers for a more detailed analysis of @xmath16 ) , where the dominant contribution is the linear one . considering also the slope of the population of sources following equation ( 26 ) , @xmath16 could be estimated from the observed autocorrelation of faint galaxies . nevertheless , measuring @xmath91 should be more feasible from the practical point of view , as we do not have to disentangle the contribution caused by lensing from the intrinsic correlations ; it is usually assumed ( and strongly hoped ) that the intrinsic ellipticities of background galaxies are not correlated . we have obtained the expressions for the correlations between the magnification or polarization of background sources and the foreground matter distribution as function of the nonlinear evolution of the power spectrum . these formuli are valid for flat and open universes .    for the crosscorrelation of the background magnification and foreground matter distribution , @xmath119 , the maximum is at zero lag and the amplitude remains above a few percent for scales @xmath122 . @xmath119 increases significantly when @xmath0 decreases . the linear contribution is negligible compared to the nonlinear one for the relevant scales below a few arcmin . varying the redshift of the foreground population , @xmath126 , a maximum amplitude of a few percent for the integrated correlation @xmath125 is obtained at @xmath151 for all background populations and all cosmological models . the crosscorrelation of the background polarization and foreground matter distribution , @xmath135 , presents a maximum of the order of @xmath142 at a non - null angle , typically in the range @xmath152 . fixing the redshifts @xmath126 and @xmath79 of the two populations , the angular scale of the maximum decreases with @xmath0 . @xmath135 increases significantly when @xmath0 decreases . the linear contribution is negligible compared to the nonlinear one for the relevant scales below a few arcmin . varying the redshift of the foreground population , @xmath126 , a maximum amplitude of @xmath153 for the integrated correlation @xmath141 is obtained within a relatively wide range @xmath154 . this amplitude grows appreciably with @xmath79 being a factor of @xmath144 difference between @xmath145 and @xmath146 .    finally , the correlation of the magnifications for a single population , @xmath149 , has a relatively small maximum amplitude @xmath155 in all cases . the amplitude grows with @xmath0 , contrary to the crosscorrelations @xmath119 , @xmath135 . the authors thank matthias bartelmann , klaus dolag and peter schneider for interesting and useful discussions . we also thank jos revuelta for his help with the aesthetic aspects of this paper . jls , emg and nbl acknowledge financial support from the spanish dges , project pb95 - 0041 . nb acknowledges a spanish m.e.c . ph.d . scholarship . some equations ( e.g. equation ( 11 ) ) have been obtained assuming the limber approximation , i.e. one assumes that the maximum scale of appreciable correlation is small compared to the typical distances to the foreground and background objects in the samples . so , if we consider a double integral with the same form as equation ( 10 ) , the contribution to the integral is appreciable only when the points are nearly at the same time @xmath156 and their separation is small , @xmath157 , when @xmath158 , the spatial separation of two points is            where @xmath163 and @xmath164 , the latter being a smooth function . the previous integral for the variable @xmath165 can be approximated by @xmath166 if one assumes that the foreground sources are placed at distances far away from the observer and hubble distance ( case of practical interest ) . finally , changing the variable @xmath165 by @xmath63 one gets equation ( 11 ) . we remark that equation ( 11 ) is also valid when a dirac distribution is assumed to represent the foreground redshift distribution . the limber approximation in fourier space has been also considered by kaiser ( 1992 ) in the context of weak gravitational lensing .
correlations between the magnification or polarization of background sources , induced by gravitational lensing due to the large - scale structure , and the positions of foreground galaxies are investigated . 
 we found that their amplitude is enhanced with respect to correlations for a single population . 
 we analize the dependence of the correlations with the density parameter @xmath0 considering a nonlinear evolution of the matter power spectrum . 
 the contribution of the linear evolution is negligible at scales below several arcminutes . 
 detailed results on the dependence of the correlations on the redshift of the foreground and background populations for different cosmological models are presented . 
 the effect reaches its maximum amplitude for foreground populations with relatively small redshifts due to the fast increase of the nonlinear matter power spectrum at recent times . 
 cosmology : theory - gravitational lensing - large - scale structure