document: the effects of weak gravitational lensing by the large - scale structure have been detected in several samples of high redshift qsos , intermediate redshift galaxies , and batse grbs . in the case of point sources , qsos and grbs , 
   weak lensing manifests itself as angular ( anti-)correlations between these sources and foreground inhomogeneously distributed mass @xcite , while in the case of galaxies weak lensing is detected through its coherent shear effect ( see @xcite for a recent review ) . in principle 
   , there is another , more direct way of detecting weak lensing , which uses fluxes of standard candles . 
   if the observed magnitudes of standard candles are corrected for cosmological distances then the effect of lensing can be seen : brighter sources will lie behind regions of mass density excess , while fainter ones will have mass deficits in their foregrounds . 
   the best example of cosmological standard candle , supernovae type ia ( snia ) have been extensively observed with the purpose of determining the global geometry of the universe @xcite . 
   nuisance effects like evolution , variations in individual sn , and gray dust extinction have been studied theoretically and observationally , and have either been corrected for or shown to be small . 
   weak lensing , another nuisance effect has been addressed theoretically by several authors @xcite and found to be unimportant given the current uncertainties . 
   for example ,  @xcite used ray tracing through cosmological simulations and found that the lensing induced dispersions on truly standard candles are @xmath3 and @xmath4 mag at redshift @xmath5 and @xmath6 , respectively , in a cobe - normalized cold dark matter universe with @xmath7 , @xmath8 , @xmath9km / s / mpc and @xmath10 . 
   these are small variations compared to the current errors which are @xmath11 mag . 
   even though weak lensing effects are estimated to be small for @xmath12 , they are predicted to be non - negligible for higher redshift sources , so it is not surprising that the highest redshift snia , sn1997ff at @xmath13 has been examined by several authors @xcite for the effects of weak lensing due to galaxies along the line of sight . 
   present day high-@xmath0 snia samples are dominated by lower redshift sne , and so have not been examined for the effects of lensing . 
   the main goal of this work is to determine if the observed fluxes of the cosmologically distant snia have suffered significantly from lensing induced ( de- ) amplifications . 
   the largest homogeneous compilation of snia has been recently published by  @xcite : table 15 of that paper contains 74 sne at @xmath14 . 
   the authors use four different light curve fitting methods ( mlcs , @xmath15 , modified dm15 , and bayesian adapted template match ) to estimate distances to sne . 
   the final quoted distance is the median of the estimates of the four individual methods , and the uncertainty is the median of the error of the contributing methods . 
   the analysis presented in @xcite yields values of the global cosmological parameters ; if a flat model is assumed , then @xmath16 and @xmath17 . 
   we use these values in all the analysis of the present paper .    as tracers of foreground mass density we use apm galaxies @xcite . 
   apm provides near full coverage of the sky in the northern and southern hemispheres , at @xmath18 . in our analysis 
   we use only the central @xmath19 of apm plates . since the plate centres are separated by @xmath20 , 
   there exist small portions of the sky that are not covered by any plate . as 
   a result of these cuts , only 55 of the 74 sne lie on the usable parts of apm plates . 
   the median redshift of the 55 sne is @xmath21 . 
   is not in our sample : it fell in the cracks between the apm plates . ] 
   since most of the sne have rather low redshifts , care must be taken to ensure that galaxies are foreground to the sne . furthermore , because sne span a large range of nearby redshifts , from @xmath22 to @xmath23 , the optimal lens redshift @xmath24 will depend on @xmath25 much more compared to a typical case where sources ( say , qsos ) are at @xmath26 and so the redshift of optimal lenses is roughly independent of @xmath25 . in our analysis 
   we adjust @xmath24 for each sn source by selecting the appropriate limiting apparent magnitude , mag@xmath27 for apm galaxies on red plates . 
   @xcite gives an empirical expression for the median redshift @xmath28 of a galaxy sample with a given faint magnitude flux cutoff . 
   this median redshift can be equated with the optimal lens redshift @xmath24 , and hence the magnitude limit of the foreground galaxies can be determined for every sn separately . however , there is a small catch . for @xmath29 optimal @xmath30 . 
   the galaxy redshift distribution whose median redshift @xmath31 has a considerable tail extending beyond @xmath32 . to avoid the problem of source / lens redshift overlap we use @xmath33 , where factor of 2 was chosen arbitrarily . 
   we explore the dependence of the results on this factor in section  [ robust ] . 
   around every sn we draw a circle of radius @xmath34 , and count the number of galaxies , @xmath35 , in the appropriate magnitude range . 
   this number is compared to the average number density in control circles , @xmath36 . 
   fractional galaxy excess is @xmath37 . 
   control circles are confined to the same apm plate as the sn , and to the same distance from the plate centre as the sn ( to reduce the effects of vignetting ) ; however , scattering the control circles randomly on the plate does not change the results significantly . for each 
   sn we also calculate @xmath38 , where @xmath39 is the number of control circles , out of total @xmath40 , that have less galaxies in them than the circle around the sn . 
   in other words , @xmath38 is the rank of the sn circle among its control ` peers ' . if sne are randomly distributed with respect to the foreground galaxies , then average @xmath41 . 
   if sne have an excess ( deficit ) of galaxies in front of them then their @xmath42 will be greater ( less ) than 0.5 . 
   analogous to the medians being more stable than averages , @xmath38 rank statistic is more stable than @xmath43 .    , and @xmath17 , and extinction , k - corrected distances from @xcite . 
   @xmath44 is a measure of the number density of foreground galaxies in circles of radius @xmath34 around sne ( see section  [ anal ] for details ) . 
   there are a total of 55 sources ( with magnitude errors ) , but only 50 ( filled points ) are used in the analysis . 
   the 50 points are grouped into three bins , whose horizontal size is shown as thick horizontal `` error - bars '' . 
   the corresponding vertical error - bars show the deviation of the mean of the points in each bin ( the rms is 4 times larger ) , and the intersection of the thick lines are the averages of the mag@xmath45 of sne in each bin . 
   the thin slanting line is the best - fit to 50 sne , and has a slope @xmath46 . 
   the significance of the correlation is @xmath2.,width=317 ]    figure  [ figone ] shows absolute magnitudes , @xmath47 , and @xmath38 ranks of 55 sne found on apm plates . 
   the effect of flux dimming due to cosmological distances has been taken out , i.e. all the sne have been ` brought ' to the same redshift ; the constant magnitude offset on the vertical axis is irrelevant for this work . 
   there are two sne whose magnitudes make them @xmath48 outliers , sn1997o , and sn1997bd represented by empty circles with dotted line error - bars . 
   we exclude these from our analysis . as quoted in @xcite . 
   sn1997o and sn1994h were excluded from the primary fit ( fit c ) by the analysis of @xcite . 
   we do not exclude sn1994h from our analysis , but if we did it would improve the trend , as its coordinates on fig . 
   [ figone ] are ( 0.427 ; -0.606 ) . ]    . 
   distance errors , @xmath49 were derived by @xcite , and listed in column 9 of their table 15 . the solid histogram is for 55 sn ; the dashed line represents 31 sn from the high-@xmath0 supernova search team . 
   the 3 outlier sne with @xmath50 are not used in the present analysis.,width=317 ]    the distance ( or , magnitude ) errors , as estimated by @xcite for the 55 sne are shown in fig . 
   [ figthree ] , as the solid line histogram . 
   the dashed line represents the subsample of 31 sne from high-@xmath0 supernova search team @xcite , whose errors appear to be generally smaller than those of the supernova cosmology project @xcite . 
   we use sne from both the groups , but exclude three outliers in fig . 
   [ figthree ] , whose @xmath51 . in fig . 
   [ figone ] these are represented by empty circles with dashed line error - bars . 
   thus , we exclude a total of 5 sne from our analysis , leaving us with 50 , shown as the solid points in fig . 
   [ figone ] . 
   these 50 sne exhibit a relation between their @xmath47 and @xmath38 , in the sense that brighter sne have an excess of galaxies in their foregrounds . for illustration purpose 
   only , we bin the 50 sne into three bins ; fig . 
   [ figone ] shows the extent of the bins and the deviation of the mean of the sne magnitudes in each bin as thick lines .    the best - fit line to the 50 sne in fig . 
   [ figone ] has a slope @xmath46 , and is shown as a thin slanting line . 
   this fit does not include magnitude errors . 
   to include the errors we do the following . 
   we calculate the best - fit slope for 10,000 realizations of the data , where each data point s sn magnitude is replaced by a randomly picked magnitude from a gaussian distribution centred on the actual magnitude value , and having width equal to the quoted error . 
   this procedure correctly incorporates the information contained in the errors , and produces a distribution of best - fit slopes , which is shown as a solid line histogram in fig . 
   [ figextra ] . 
   this distribution shows that the median best - fit slope is @xmath52 , while @xmath53 ( i.e. a case of no correlation ) is ruled out at 99.80% confidence level . 
   had we used 53 sne ( i.e. had we not exluded the 3 sne with large distance errors ) , the median best - fit slope would gave been @xmath54 , while @xmath53 would have been ruled out at 99.37% confidence level . 
   the corresponding distribution of @xmath55 slopes is shown as the dashed line in fig . 
   [ figextra ] .    ? 
   the histograms show the distribution of derived @xmath55 values in 10,000 realizations of the data . 
   the solid ( dashed ) line histogram represents results using 50 ( 53 ) sne . 
   the short vertical lines at the bottom of the plot indicate the medians of the distributions.,width=317 ]    next , we determine the likelihood of this relation arising by chance . to that end 
   , we estimate the significance of the relation in two ways . 
   first , we assign random positions to sne ( keeping these control sne on the same apm plate as the original source ) , and redo all the analysis . 
   we repeat this 1000 times , and in 8 cases we find @xmath56 , implying statistical significance of @xmath57 . 
   second , we take the list of observed @xmath47 values and randomly reassign them to observed sn sky positions . 
   10,000 randomized sne samples are created in this fashion , and only @xmath58 of these have @xmath59 . 
   based on these two tests we conclude that the significance of the @xmath47@xmath38 relation is better than @xmath60 . 
   these results are consistent with weak gravitational lensing , which would amplify sne found behind more nearby mass concentrations , as traced by apm galaxies . 
   alternatively , the results could be due to the action of galactic dust , which will obscure certain directions of the sky making galaxies less numerous and sne fainter . 
   we consider galactic dust further in section  [ disc ] ; in sections  [ model ] and  [ montecarlo ] we proceed on the assumption that weak lensing is responsible to the @xmath47@xmath38 relation . 
   the distribution of sne points in fig .  [ figone ] depends on specific choices that we made for certain parameters , in particular we chose circles of radius @xmath34 and galaxy magnitude limit such that @xmath61 , where @xmath62 ( see section  [ data ] ) . 
   how would the results change if different choices were made ? 
   in other words , how robust is our result , would it disappear had we picked a different set of parameters ?     of circles around sne . 
   the vertical axis shows @xmath55 , the slope of the best - fit line to the relation of the type shown in fig  [ figone ] . 
   the filled circles represent cases where galaxies were counted around sn ; stars symbols represent cases where galactic stars were used instead ( i.e. control experiment ) . 
   the dotted lines denote , _ approximately _ , the levels of statistical confidence . 
   the dashed vertical line indicates radius @xmath63 used in the analysis of section  [ anal].,width=317 ]    figure  [ figthetatest ] shows the effect of changing @xmath63 . 
   the vertical axis is @xmath55 , the best - fit to the @xmath47@xmath38 relation in each case . 
   filled points represent cases where _ galaxies _ were counted and used to determine the @xmath38 rank , while star symbols represent cases where _ galactic stars _ were used instead . 
   as expected , @xmath64 do not correlate with sne magnitudes , and the values of the best - fit slopes are near zero . 
   however , because the apm star - galaxy classifier is not perfect , some ` stars ' are actually galaxies , which accounts for some signal being seen when using @xmath64 . 
   horizontal dotted lines mark the approximate location of @xmath65 , and @xmath60 confidence levels . 
   these are only approximate because every point in the plot will have its own significance level , but because the number of sne contributing to each point is the same in each case , and the total dispersion in sne magnitudes is the same , same @xmath55 values have about the same significance , regardless of @xmath63 . 
   we note that for very small @xmath63 the galaxy numbers become very small , and poisson noise drowns out any @xmath47@xmath38 correlation that might exist , so the upturn in the values of the best - fit slope at small @xmath63 is probably not real . 
   the dashed vertical line marks the @xmath63 value used in section  [ anal ] . 
   we conclude that significant @xmath47@xmath38 anti - correlations occur only with galaxies and not with galactic stars ( which serve as a control sample ) , and only for @xmath66 .    . 
   similar to fig . 
   [ figthetatest ] ; see section  [ robust ] for details.,width=317 ]    figure  [ figmagtest ] shows the effect of changing the median redshift , @xmath61 of the apm galaxies , or equivalently , mag@xmath27 . 
   the hidden variable which is varied along the horizontal axis is @xmath67 . 
   since each sn has a different value of @xmath24 and hence mag@xmath27 , depending on its @xmath25 , there is no unique way of labeling the horizontal axis by using galaxy magnitudes , or redshifts . 
   we label that axis by assuming @xmath68 , the median of the sne redshift distribution . at the top of the plot 
   we show how galaxy magnitude limit on the horizontal axis translates into the median galaxy redshift ( for a specific case of @xmath68 ) . 
   we see that significant anti - correlation between @xmath47 and @xmath38 of foreground galaxies occurs for galaxies with @xmath69 . using galactic stars instead of galaxies 
   produces no significant results . 
   the value we used in section  [ anal ] is shown with a vertical dashed line , and corresponds to @xmath70 . as in fig . 
   [ figthetatest ] the upturn in the best - fit slope values at bright mag@xmath27 is probably due to poisson noise . the range of redshifts of apm galaxies that act as the best lenses for sne are @xmath71 ; more distant galaxies show no signal , as expected , if lensing is the correct interpretation of the data , because more distant galaxies are either too close to sne in redshift or are actually at the same @xmath0 . 
   a number of other tests have been carried out as well . 
   for example , instead of using a @xmath25-dependent mag@xmath27 , we tried fixed values of @xmath72 , which gave , @xmath73 , respectively , comparable to , but somewhat smaller than those seen in fig . 
   [ figmagtest ] . 
   this is not surprising : @xmath25-dependent galaxy magnitude limits pick optimal lens redshifts for each source , thus maximizing the observed lensing signature . 
   we also reran the analysis with subsamples of the entire 50-source sample . 
   we split the sne according to the teams : 20 supernova cosmology project snia gave a @xmath47@xmath38 anti - correlation significant at 95.8% , while the corresponding significance level for the 30 sne from the high-@xmath0 supernova search team is 96% . because the quality of ukst apm ( southern hemisphere ) plates is higher than poss apm ( northern hemisphere ) plates , and because there is some overlap between plates along the equator , we used ukst plates whenever possible . redoing the analysis using only the 36 ukst sne we get the best fit slope @xmath74 at a significance level of 99.6% , 
   while the 14 poss sne had @xmath75 and the correlation was not significant , which is not surprising given the size of this subsample . splitting the whole sample into low and high redshift groups we get the following results : 25 sne with @xmath76 
   have a slope @xmath77 at 96.8% , while 25 sne with with @xmath78 have a slope @xmath79 at 88.9% . 
   these tests suggest that the @xmath47@xmath38 relation has a physical origin ( weak lensing or galactic dust ) , and is not an artifact arising form one subset of the data . 
   in this section we adopt the weak lensing interpretation of the @xmath47@xmath38 relation . 
   our simple lensing model does not use the individual values of @xmath25 , and the optimal redshift distribution of the corresponding lenses . in lieu of these parameters 
   we use the lensing optical depth of the matter traced by the galaxies , @xmath80 . for a fixed global geometry 
   , @xmath80 depends on @xmath25 and the redshift distribution of the mass traced by the galaxies . 
   we assume that the apm galaxies faithfully trace the mass up to some redshift , @xmath81 , then @xmath82^{1/2}\ ; \sigma_{crit}(z , z_s ) } }           \ ; dz,\end{aligned}\ ] ] in that case , projected fractional mass excess is @xmath83 , where @xmath84 is the convergence , with respect to a smooth universe ( filled beam ) for the corresponding line of sight . in the weak lensing regime source amplification @xmath85 ; @xmath86 for an unlensed source . the smallest value that @xmath84 can attain is minus the total optical depth , @xmath87 , which corresponds to emptying out all the mass along the line of sight from the observer to @xmath81 .    in addition to the value for @xmath80 , our model has three ingredients , ( 1 ) mass distribution , i.e. a probability distribution ( pdf ) for @xmath88 , which is related to @xmath84 pdf , ( 2 ) a biasing scheme , which relates @xmath88 to the projected fractional galaxy number density excess , @xmath43 , and ( 3 ) the dispersion in the intrinsic magnitudes of snia s . 
   given this information we generate synthetic snia samples , with 50 sources each , then compute @xmath38 each sn , and the observed sn magnitude , @xmath47 . 
   the slope of the best - fit line to the @xmath47@xmath38 relation , @xmath55 is then used to test how well a given model reproduces the observations .    because our model assumes the same @xmath24 distribution for all sources , the value of @xmath55 to compare our models predictions to should be the one obtained by using a constant mag@xmath89 , i.e. @xmath90 ( see fourth paragraph in section  [ robust ] ) . 
   the specifics of the three model ingredients , and the associated model parameters are described later in this section . 
   the goal of the modeling is to determine what set of parameters can reproduce the observations , i.e. have @xmath91 . 
   we are particularly interested in what effect biasing has on the results . 
   it is often suggested that the large amplitude of qso - galaxy correlations mentioned in the introduction is , at least in part , due to the fact that biasing is not a simple linear , one - to - one mapping from @xmath88 to @xmath43 . 
   qso - galaxy correlation function ( @xmath92 ) , galaxy autocorrelation ( @xmath93 ) and the matter fluctuation power spectrum ( @xmath94 ) are related by @xmath95 where @xmath96 is the biasing parameter . 
   this means that @xmath92 probes a _ combination _ of @xmath96 and @xmath94 . 
   thus , for a given @xmath94 , @xmath92 can be enhanced with positive biasing , especially if it is non - linear on relevant spatial scales . 
   the important difference between weak lensing induced qso - galaxy correlations and the effect on standard candles we have studied here is that in the latter case biasing plays a minor role . 
   the slope of the @xmath47@xmath38 relation is most sensitive to the mass distribution . 
   our modeling demonstrates this later ; first , we described the specifics of the three model ingredients . 
   based on the results from cosmological n - body simulations @xcite the shape of the probability distribution function ( pdf ) of @xmath84 is very roughly gaussian , but asymmetric , with the most probable @xmath84 , which we call @xmath97 being less than 0 , i.e. most sources are deamplified . 
   the tail of the @xmath84 distribution extends to high positive values of @xmath84 . 
   we use these published results of @xcite to construct an approximate shape for our @xmath84 pdf : @xmath98 ^ 2/2\sigma_{\kappa_1 } ) , \quad { \rm for}\quad \kappa > \kappa_m\\ exp\;(-[\kappa-\kappa_m]^2/2\sigma_{\kappa_2 } ) , \quad { \rm for}\quad \kappa\le\kappa_m \end{array } \right . 
   \label{stdpdf}\ ] ] the pdf is a combination of two half - gaussians , with two widths , @xmath99 and @xmath100 , which describe the high @xmath84 and the low @xmath84 sides of the pdf respectively . 
   the location of the peak of the pdf , @xmath97 is adjusted such that the average @xmath84 is 0 , which implies @xmath101 @xmath97 is always negative , since the skewness of the pdf dictates that @xmath102 . from the numerically computed pdfs of @xcite we estimate that for a flat @xmath103 model , where the sources are at @xmath104 , and the smoothing scale is @xmath105 , the following approximations apply : @xmath106 , and @xmath107 . 
   thus we now have a realistic @xmath84 pdf , appropriate for a standard cosmology with standard mass distribution . 
   figure  [ grid](d ) shows such a pdf as a short - dash line . 
   in addition to the standard @xmath84 pdf described above , we also try an extreme form for @xmath84 pdf obtained using the `` bifocal lens '' mass distribution proposed by @xcite : @xmath108 where @xmath109 is the kronicker s delta function , @xmath110 , and @xmath111 . for a fixed @xmath80 , 
   this pdf can produce more pronounced lensing effects than the standard pdf , if @xmath112 is assigned the smallest allowable value , and @xmath113 is made very large . 
   this is because the standard pdf has a large probability near @xmath86 , while the bifocal pdf avoids such values altogether .    the distribution of mass corresponding to eq . 
   [ bifocals ] is unrealistic . 
   one can make it somewhat realistic by allowing a range of @xmath113 and @xmath112 values . 
   we do the following : for any one line of sight we randomly pick @xmath113 and @xmath112 from specified ranges , using flat priors , and eq . 
   [ bifocals ] sets the probability distribution of the two amplifications . 
   we use two sets of ranges : ( 1 ) @xmath114 and @xmath115 ( we will call this model bifocal i model ) , ( 2 ) @xmath116 and @xmath117 ( we will call this bifocal ii ) . the corresponding @xmath84 pdfs , obtained by considering many lines of sight are shown in fig . 
   [ grid](d ) as a solid line for bifocal ii and as a long - dash line for bifocal i. in bifocal ii the minimum value of @xmath112 is the minimum allowed value ; the other limits on @xmath118 s were picked arbitrarily . for comparison , 
   the standard @xmath84 pdf of section  [ stdmass ] is also shown , as the short - dash line . 
   optical depth of @xmath119 was assumed for all three . 
   compared to a standard pdf , bifocal pdfs imply that most of the lines of sight are rather empty , and there are a few lines of sight with very high values of @xmath84 . 
   projected fractional mass excess , @xmath88 is related to the observationally accessible quantity , @xmath43 through biasing . 
   we chose `` power law '' biasing , motivated by numerical simulations of @xcite and @xcite , @xmath120 where the power law part allows the biasing to be non - linear , with different indexes depending on the sign of @xmath88 , while @xmath121 is a stochastic biasing component which is chosen randomly for each sn ; @xmath121 distribution has a gaussian shape and width @xmath122 . 
   factor @xmath123 multiplying @xmath121 ensures that the dispersion in @xmath43 is reduced in underdense regions . 
   qualitatively , the @xmath43 vs. @xmath88 relations produced by eq . 
   [ biasing ] look similar to those in fig.1 of @xcite . 
   we assume that the dispersion in magnitude about the perfect standard candle case has a gaussian shape , with width @xmath124 . 
   each model has five independent parameters : + @xmath125 . we assume flat priors for the three biasing parameters , @xmath126 , @xmath127 , and @xmath122 , as well as for @xmath124 . 
   each specific set of parameters together with one of the three mass distribution models generates a synthetic realization of the 50-source snia sample . from this entire ensemble of realizations , we only consider those that satisfy these observational constraints : ( 1 ) the total rms dispersion in the synthetic snia magnitudes ( which includes intrinsic and lensing induced contributions ) must be within 0.05 mag of the actual observed value of 0.3mag ; ( 2 ) the moments of the synthetic @xmath43 distribution ( average , standard deviation , skewness and kurtosis ) must be reasonably close to those of the observed distribution , which is characterized by ( 0.063 , 0.52 , 1.7 , 4.9 ) . 
   we chose `` reasonably close '' to mean that synthetic values must not be more than a factor of 2 away from the actual values . because of these constraints , the range from which the values of @xmath126 , @xmath127 , @xmath122 , and @xmath124 parameters are picked is not relevant , as long as it not too restrictive . 
   in other words , the observational constraints eliminate cases with very large values of these parameters . 
   the data suggests that demagnified sne are not lost due to the flux limit . if they were , we would have less sne at small values of @xmath38 , and more at high values of @xmath38 , whereas in the data ( fig .  [ figone ] ) 
   the sources are roughly equally spread over the entire @xmath38 range . 
   so our synthetic lensing models assume that we do not lose sne because of deamplification .     of the @xmath47 vs. @xmath44 relation ) given different mass distribution models , and a range of optical depths . 
   solid dots assume standard @xmath84 pdf of section  [ stdmass ] , empty squares and star symbols represent bifocal i and ii models , respectively , discussed in section  [ bifocalmass ] . 
   each point was obtained using 10,000 synthetic realizations of snia 50-source sample . 
   see section  [ montecarlo ] for details.,width=317 ]    for each synthetic realization of the 50-source snia sample we derive the corresponding @xmath47@xmath38 relation . the slope of the best - fit line , @xmath55 is recorded . for a given value of @xmath80 
   we generate 10,000 realizations . 
   solid dots in fig . 
   [ figoptdepth ] show the percentage of realizations that have @xmath55 smaller than the observed value of -0.3 , as a function of total optical depth , for the standard @xmath84 pdf . 
   empty squares and star symbols represent the results for the bifocal i and ii . 
   figure  [ figoptdepth ] considers a range of optical depths . 
   what is the appropriate value for @xmath80 ? 
   if the source is at the median source redshift for our sample , @xmath68 , then apm galaxies sample the mass distribution up to a redshift of about 0.3 . in this case 
   the lensing optical depth probed by apm galaxies is 0.011 ( left - most arrow in fig . 
   [ figoptdepth ] ) . a somewhat more optimistic estimate for @xmath80 
   is obtained if we assume that galaxies trace the mass fluctuations up to the median source redshift , 0.47 , which gives @xmath128 ( middle arrow ) . 
   the limiting case is obtained by considering the most distant source , at @xmath129 , and assuming that galaxies probe mass up to a redshift of 0.47 , which gives @xmath130 ( right - most arrow ) .    the important conclusion from this figure is that if the standard mass distribution ( section  [ stdmass ] ) is assumed , then the value of @xmath80 does not really matter , and the probability of reproducing the observations is @xmath131 , if @xmath80 is within a reasonable range . if , on the other hand , the bifocal pdf i or ii are assumed , the probability of reproducing observations depends sensitively on the assumed optical depth . if @xmath130 then these models yield 6 - 20% probability . 
   the bifocal pdfs produce more discernible lensing effects , for the same @xmath80 because they have a wider range in @xmath118 , as seen in fig . 
   [ grid](d ) .    @xmath44 relation on the three biasing parameters ( see section  [ biasscheme ] ) . 
   short - dash lines are for the standard @xmath84 pdf model of section  [ stdmass ] , while the solid lines are for the bifocal ii @xmath84 pdf described in section  [ bifocalmass ] . 
   both models assume optical depth @xmath132 . 
   the contours are based on 50,000 realizations , and are spaced a factor of 3 apart . 
   the shapes of the contours are partly determined by the observational constraints , described in section  [ montecarlo ] ( second paragraph ) . 
   it is evident that biasing parameters have little impact on the @xmath55 . 
   _ panel ( d ) : _   short - dash and solid lines represent @xmath84 pdfs corresponding to the contours in panels ( a)-(c ) of the same line type . 
   the minimum @xmath84 in the bifocal ii pdf is @xmath87 ( indicated by an arrow ) . 
   the long - dash line represents bifocal i @xmath84 pdf . 
   , width=317 ]    the results presented in fig . 
   [ figoptdepth ] marginalize over values of all the model parameters ( except @xmath80 ) , so the effects of the biasing parameters on @xmath55 are hidden . 
   however , it turns out that these parameters do not correlate with @xmath55 . 
   figure  [ grid ] shows the dependence of @xmath55 on @xmath126 , @xmath127 and @xmath121 for the bifocal ii @xmath84 pdf model ( solid contour lines ) , and the standard @xmath84 pdf model ( dashed contour lines ) , both for @xmath132 . 
   fifty thousand realizations were created for each model , and the corresponding contour lines plotted , with adjacent contours separated by a factor of 3 . 
   it is apparent that for both models the dependence on the biasing parameters is weak . 
   we conclude that _ the slope of the @xmath47@xmath44 relation depends on the amount and distribution of matter , i.e. on @xmath80 and the shape of the @xmath84 pdf model , but does not depend on the specifics of the biasing scheme . 
   _ figure  [ figoptdepth ] quantifies the dependence on amount and distribution of mass : as expected , higher optical depth results in more pronounced lensing effects . 
   standard @xmath84 pdf produces less lensing for the same @xmath80 compared to a much broader bifocal pdf . 
   figure  [ grid ] demonstrates that biasing has little effect on the slope of @xmath47@xmath44 relation . in other words , the type of lensing signature considered here ( @xmath47@xmath44 relation for standard candles ) probes mass distribution , and is independent of biasing . 
   we detect the signature of weak lensing in the current sample of 50 high redshift snia taken from two teams : high-@xmath0 supernova search team and supernova cosmology project . after correcting sne magnitudes for cosmological distances ( assuming @xcite values ) 
   , we find that brighter sne are preferentially found behind regions overdense in foreground galaxies . 
   this @xmath47@xmath44 relation has a slope @xmath133 to @xmath134 , when the angular radii of foreground regions are 5 - 15 arcmin . 
   the statistical significance is @xmath2 ( see fig .  [ figthetatest ] and  [ figmagtest ] ) . 
   the angular radii of 5 - 15 arcmin imply that the lensing structures are @xmath135 mpc across if they are located at a redshift of 0.1 , and so correspond to non - linearly evolved intermediate - scale structure .    aside from the possibility that the observed @xmath47@xmath44 relation is a fluke , 
   there is one other possible , non - lensing interpretation : galactic dust obscuration , which would make snia sources brighter and apm galaxies more numerous in the directions devoid of galactic dust . 
   if this interpretation is correct , then ( 1 ) galactic extinction corrections applied to the sne @xcite are much too small , and ( 2 ) galactic dust is able to change projected apm galaxy number density by @xmath136 , since the observed rms in @xmath43 is @xmath137 . 
   we can not comment on ( 1 ) , but can estimate the effect of ( 2 ) . 
   how much excess / deficit in galaxy density can galactic dust create ? using @xcite data we calculate the average @xmath138 galactic extinction for the 50 sne sample to be 0.18 mag . 
   the slope of the r - band apm galaxy number counts is @xmath139 , therefore typical `` dust - induced '' @xmath140 . 
   this is an upper bound because extinction in the @xmath141 band will be smaller than in @xmath138 , and because extinction averaged over 5 - 15 arcmin radius patches will be smaller than estimates for individual sne , given that @xcite extinction map resolution is 6@xmath142.1 fwhm . if galactic dust is indeed the cause of the @xmath47@xmath44 correlation , then @xmath47 and @xmath44 should separately correlate with sne extinction . in reality 
   , @xcite extinction estimates do not show a correlation with either @xmath47 or @xmath44 . 
   overall , galactic dust interpretation of the observed @xmath47@xmath44 relation seems unlikely , but can not be ruled out .    note that dust _ 
   intrinsic _ to the @xmath143 structures probed by the apm galaxies can not be invoked to explain the observations , because such dust would produce an effect opposite to the one detected here , i.e. @xmath55 would be positive . 
   if dust is present in groups and clusters traced by the apm , it will diminish the amplitude of the effect we detect . 
   presence of dust in groups was suggested by @xcite who found that faint qso candidates are anti - correlated with foreground groups . 
   however , such an anti - correlation can also be explained by weak lensing @xcite . 
   current observations indicate that groups and clusters do not contain significant amounts of dust @xcite . 
   if the lensing interpretation in correct , then the standard models of mass distribution have some difficulty in reproducing @xmath55 ; the observed value would be detected only in @xmath131 of the cases . 
   we investigate how @xmath55 is affected by the amount and distribution of mass along the light of sight to the sources , and galaxy biasing schemes . 
   we find that larger optical depths and broader @xmath84 pdf result in steeper @xmath55 slopes ( fig . 
   [ figoptdepth ] ) . 
   optical depth is a function of global geometry , and no realistic cosmological model can give @xmath144 , which is what would be required to comfortably explain the results . 
   broader @xmath84 pdfs mean that mass fluctuations are more extreme than the standard cosmological models allow , a scenario which is in apparant conflict with other means of determining mass fluctuations , like cosmic velocity flows and weak shear lensing . 
   we also find that biasing has little effect on @xmath55 ( fig . 
   [ grid ] ) . 
   this insensitivity to biasing is in contrast to weak lensing induced qso - galaxy and grb - galaxy ( anti-)correlations , where biasing could , at least partly explain the higher than expected amplitude of the effect @xcite . 
   we conclude that weak lensing of standard candles provides a cleaner probe of the mass distribution at @xmath145 , on @xmath146few mpc scales , then lensing induced angular correlations . 
   in fact , snia can provide the perfect means of measuring mass inhomogeneities using gravitational lensing . 
   a set of standard candles at known redshifts can be analyzed using the complementary techniques of weak magnification and weak shear lensing . 
   the advantage of magnification lensing over the more commonly used shear lensing is that with the former one can chose the redshift range of the lenses , whereas the latter yields the cumulative effect of lensing along the entire line of sight to the source . 
   therefore a large uniform set of intermediate redshift snia , such as the ones that would result from the snap mission and the lsst missionhome.html ] would be invaluable for the studies of mass clustering in the nearby universe . 
   in addition to detecting intermediate and high - redshift sne for the purposes of estimating the global cosmological parameters and the equation of state of the dark energy , snap will also measure weak shear lensing signature due to large scale structure . combining magnification data ( of the type considered in this paper ) and shear information @xcite for 
   a large set of sne will allow the study of mass distribution at @xmath147 with unprecedented accuracy and 3-dimensional spatial resolution . 
   the final issue we address is the impact of weak lensing on the determination of global cosmological parameters , @xmath148 and @xmath149 using snia standard candles . in principle 
   , weak lensing can affect the derived values of @xmath148 and @xmath149 , if ( 1 ) demagnified sne are preferentially lost from the sample due to faint flux cutoff , and/or ( 2 ) the @xmath84 pdf is asymmetric and the sn sample size is small ( see also @xcite ) . 
   if demagnified sne were lost from the sample then the distribution of sne in @xmath44 would be skewed in the direction of larger @xmath44 values . 
   if @xmath84 pdf is asymmetric , ( and it is , on scales considered here ) , then most sne in a small sample will be slightly demagnified compared to average , and will have their @xmath44 skewed in the direction of smaller values . 
   so both ( 1 ) and ( 2 ) would make the distribution of sne in @xmath44 uneven , and would to some extent cancel each other . in the present sample of 50 
   , the distribution of sne in @xmath44 is indistinguishable from uniform , so the average @xmath47 corresponds to the average @xmath44 , and hence lensing effects by @xmath146mpc - size structures probably did not bias the derived values of @xmath148 and @xmath149 . 
   the authors would like to thank peter garnavich and jason rhodes for their careful reading of the manuscript and valuable comments . 
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summary: the current sample of high - redshift supernova type ia , which combines results from two teams , high-@xmath0 supernova search team and supernova cosmology project , is analyzed for the effects of weak lensing . after correcting sne magnitudes for cosmological distances , assuming recently published , homogeneous distance and error estimates , we find that brighter sne are preferentially found behind regions ( 5 - 15 arcmin radius ) which are overdense in foreground , @xmath1 galaxies . 
   this is consistent with the interpretation that sne fluxes are magnified by foreground galaxy excess , and demagnified by foreground galaxy deficit , compared to a smooth universe case . 
   the difference between most magnified and most demagnified sne is about 0.3 - 0.4 mag . 
   the effect is significant at @xmath2 level . 
   simple modeling reveals that the slope of the relation between sn magnitude and foreground galaxy density depends on the amount and distribution of matter along the line of sight to the sources , but does not depend on the specifics of the galaxy biasing scheme . 
   [ firstpage ]    gravitational lensing 
   supernovae : general