Astronomy&
Astrophysics

A&A 403, 817–828 (2003)
DOI: 10.1051/0004-6361:20030406
c ESO 2003

Improving the accuracy of cosmic magnification statistics
B. Ménard1,2 , T. Hamana3,1 , M. Bartelmann1 , and N. Yoshida4,1
1

Max-Planck-Institut für Astrophysik, PO Box 1317, 85741 Garching, Germany
Institut d’Astrophysique de Paris, 98 bis Bld Arago, 75014, Paris, France
3
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
4
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge MA 02138, USA2

Received 4 October 2002 / Accepted 17 March 2003
Abstract. The systematic magnification of background sources by the weak gravitational-lensing effects of foreground matter,also called cosmic magnification, is becoming an efficient tool both for measuring cosmological parameters and for exploring
the distribution of galaxies relative to the dark matter. We extend here the formalism of magnification statistics by estimating
the contribution of second-order terms in the Taylor expansion of the magnification and show that the effect of these terms waspreviously underestimated. We test our analytical predictions against numerical simulations and demonstrate that including
second-order terms allows the accuracy of magnification-related statistics to be substantially improved. We also show, however,that both numerical and analytical estimates can provide only lower bounds to real correlation functions, even in the weak lensing regime. We propose to use count-in-cells estimators rather than correlation functions for measuring cosmic magnification
since they can more easily be related to correlations measured in numerical simulations.
Key words. cosmology: gravitational lensing – cosmology: large-scale structure of Universe

1. Introduction
Gravitational lensing by large-scale structures magnifies
sources and distorts their images. The systematic distortion offaint background galaxies near matter overdensities, the cosmic shear, has been measured by several groups in the past few
years (Bacon et al. 2000, 2002; Hämmerle et al. 2002; Hoekstra
et al. 2002; Kaiser et al. 2000; Maoli et al. 2001; Réfrégieret al. 2002; Rhodes et al. 2001; Van Waerbeke et al. 2000, 2001,
2002; Wittman et al. 2000). It was found to be in remarkableagreement with theoretical predictions based on the Cold Dark
Matter model, and has already provided new constraints on cosmological parameters (Van Waerbeke et al. 2001).
In a similar way, systematic magnifications of backgroundsources near foreground matter overdensities, the cosmic magnification, can be measured and can provide largely independent constraints on cosmological parameters (Ménard &Bartelmann 2002; Ménard et al. 2002). Gravitational magnification has two effects: first, the flux received from distantsources is increased, and the solid angle in which they appear
is stretched, thus their density is diluted. The net result of thesecompeting effects depends on how the loss of sources due todilution is balanced by the gain of sources due to flux magnification. Sources with flat luminosity functions, like faint galaxies, are depleted by cosmic magnification, while the number
Send offprint requests to: B. Ménard,
e-mail: menard@mpa-garching.mpg.dedensity of sources with steep luminosity functions, likequasars, is increased. Thus, cosmic magnification gives rise
to apparent angular cross-correlations between backgroundsources and foreground matter overdensities which are physically completely uncorrelated with the sources. These overdensities can be traced by using the distribution of foregroundgalaxies.
Numerous studies have demonstrated the existence ofquasar-galaxy correlations on angular scales ranging from one
arcminute to about one degree, as expected from cosmic lensing (for a review, see Bartelmann & Schneider 2001; also
Guimarães et al. 2001). In many cases, the measured correlation amplitudes have been higher than the theoretical predictions, however a number of non-detections have also been reported, leaving the true amplitude of the effect unclear from theobservational point of view.
While cosmic shear can directly be related to observable
quantities like image ellipticities, the theoretical interpretationof cosmic magnification involves several approximations:– the luminosity function of the sources is described by apower-law over the range probed by the flux limit of theobservation; and– the magnification is assumed to fall into the weak lensing
regime, i.e. to deviate weakly from unity. Thus, the magnification can with sufficient accuracy be approximated by
its first-order Taylor expansion and its deviation from unity
becomes proportional to the lensing convergence alone.

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B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

While the first assumption is comfortably satisfied, in particularfor quasars, the validity of the second needs to be verified. Thisis the goal of the present paper.
Our paper is structured as follows: first, we introduce theformalism of the effective magnification and its Taylor expansion in Sect. 2. We then describe a number of statistics related
to the lensing convergence, and evaluate the amplitude of thesecond-order terms which appear in the Taylor expansion. In
Sect. 3, we describe the numerical simulations we use to test
our analytical results and estimate the accuracy of several approximations for the magnification. As an application, we investigate second-order effects on quasar-galaxy correlations in
Sect. 4, and we summarise our results in Sect. 5.

2. Formalism

2.1. Expanding the magnification
Cosmic magnification can be measured statistically throughcharacteristic changes in the number density of the backgroundsources. Along a given line-of-sight, this effect depends on two
quantities:– the magnification factor µ, which describes whether
sources are magnified or demagnified, depending onwhether the matter along their lines-of-sight is preferentially over- or underdense compared to the mean,– and the logarithmic slope α of the source counts as afunction of flux, which quantifies the amplitude of source
number-count modifications due to flux magnification. As
mentioned in the introduction, magnification by gravitational lensing not only increases the observed flux, but also
stretches the sky, thus the number density of sources on amagnified patch of the sky is reduced. The net magnification effect, called magnification bias, depends on the balance between the number of sources lost by dilution andgained by flux magnification. The steeper the number-countfunction of the sources is, the more pronounced is the magnification bias.
If the number-count function of the background sources canbe described as a power law in a sufficiently wide range aroundthe flux limit of the observation, the magnification bias is quantified by the effective magnification µα−1 . It directly expresses
the changes of the background source density caused by lensing through the relationn(> S , θ) = µα−1 (θ) n0 (> S )(1)

where n0 (>S ) is the intrinsic number-count function of sources
whose observed flux exceeds S in the absence of lensing,and n(>S ) is the corresponding number-count function in presence of lensing.
The local properties of the gravitational lens mapping arecharacterised by the convergence κ, which is proportional to
the surface mass density projected along the line-of-sight, andthe shear γ, which is a two-component quantity and describes
the gravitational tidal field of the lensing mass distribution. The
effective magnification is related to κ and γ through
h
i1−α
,
(2)
µα−1 = (1 − κ)2 − |γ|2

where |γ| = (γ12 + γ22 )1/2 is taken as the absolute value of theshear. In the weak-lensing regime, both κ and |γ| are small compared to unity, and the previous expression can be expanded ina Taylor series:hiµα−1 = 1 + (α − 1) 2κ + (2α − 1)κ2 + |γ|2 + O κ3 , |γ|3 . (3)Previous studies using analytical formulae for magnification
statistics focused only on the first-order term of this expansion,i.e. they used the approximation µα−1 ≈ 1 + 2(α − 1) κ, which
potentially causes the amplitude of the effect to be underestimated. In this section, we investigate the second-order terms in
the expansion and estimate their contribution.
In doing so, we first note that κ2 (θ) and |γ|2 (θ) share thesame statistical properties (e.g. Blandford et al. 1991), because
both κ and γ are linear combinations of second-order derivatives of the lensing potential. The identity of their statistics ismost easily seen in Fourier space. Since we will only deal withensemble averages of the magnification later on, κ2 and |γ|2 canbe combined into a single variable, which we denote by κ forsimplicity. Thus, we can write for our purposes,hi(4)µα−1 = 1 + 2(α − 1) κ + ακ2 + O(κ3 ) .
Observable effects are due to departures from the mean value ofthe magnification. Therefore, the relevant quantity to correlateis δµα−1 = µα−1 − hµα−1 i. Then, up to second order in κ2 , theautocorrelation function of the effective magnification ishδµα−1 (φ) δµα−1 (φ + θ)i = 4(α − 1)2 hκ(φ)κ(φ + θ)ii(5)+ 2αhκ(φ)κ2 (φ + θ)i ,and the corresponding power spectrum can be expanded in asimilar way,ih(6)Pµα−1 (s) = 4(α − 1)2 Pκ (s) + 2αPµ,2 (s) ;the power spectrum Pµ,2 (s) will be defined in Eq. (16) below.
The last two equations show that the importance of the secondorder terms in the expansion (3) increases as the number-countfunction of the background sources steepens, i.e. as α increases.
In the following, we will use α = 2 unless stated otherwise.
This value applies, for instance, to the number counts of brightquasars with mB < 19.5 (Pei 1995). For simplicity, we abbreviate hδµ δµi by hµ µi.

2.2. Second and third-order correlations
We will now estimate several κ-related statistical quantities
needed in the Taylor expansion of the magnification. For thispurpose, we first introduce the κ projector such thatZ wHdw pκ (w)δ θ fK (w), w(7)
κ(θ) =
0can be written as a weighted line-of-sight projection of the density contrast δ from the observer to the Hubble distance wH . The
projector is H 230pκ (w) = Ω02ZcwH
fK (w) fK (w0 − w)
dw0
nS (w0 )
,
(8)
×
a(w)
fK (w0 )w

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

819

where w is the radial coordinate distance, fK (w) is the comoving angular-diameter distance, nS (w) is the normalised distance
distribution of the sources, and a(w) is the cosmological scale
factor. Using Limber’s equation, we can then relate the autocorrelation function of κ to the dark-matter power spectrum Pδ ,Z
p2 (w)
hκ(φ)κ(φ + θ)i =dw 2κ
fK (w)!
Zs
sds
Pδ, w J0 (s θ) ,(9)×2π
fk (w)
where J0 is the zeroth-order Bessel function, and the power
spectrum Pκ corresponding to this correlation function is!
Z
p2 (w)
s
,w ·
(10)
dw 2κ
Pδ
Pκ (s) =fk (w)
fK (w)As indicated by Eq. (5), the estimation of second-order terms
requires the computation of the cross-correlation between κand κ2 . We do this by first introducing a three-point correlation function for κ and then identifying two of its three points.
As usual, we define the three-point function byzκ (θ1 , θ2 ) = hκ(φ) κ(φ + θ1 ) κ(φ + θ2 )i .
Using the κ projector defined in (8), we can then writeZZZ
dw1 pκ (w1 ) dw2 pκ (w2 ) dw3 pκ (w3 )zκ (θ1 , θ2 ) = × δ fK (w1 )φ, w1 δ fK (w2 )(φ + θ1 ), w2
× δ fK (w3 )(φ + θ2 ), w3 .

(11)

(12)

Next, we employ the approximation underlying Limber’s equation, which asserts that the coherence length of the density fluctuation field is much smaller than the scales on which the projector pκ varies appreciably. Finally, we insert the expression
for the bispectrum of the dark-matter fluctuations detailed inAppendix A, and find
Z
Z 2
Z 2
d k1 ik1 ·θ1 fK (w)d k2
dw p3κ (w)ezκ (θ1 , θ2 ) =
(2π)2
(2π)2
× Bδ (k1 , k2 , −k1 − k2 , w) eik2 ·θ2 fK (w) ,(13)
where Bδ (k1 , k2 , k3 ) is defined byhδ̂(k1 )δ̂(k2 )δ̂(k3 )i = δD (k1 + k2 + k3 ) Bδ (k1 , k2 , k3 ) .

(14)Then, using Eq. (12) and identifying two points of the threepoint correlation function θ1 → θ2 ≡ θ (or equivalently θ1or θ2 → 0), we findZ
Z 2
d k1 ik1 ·θ fK (w)dw p3κ (w)ehκ(φ)κ2 (φ + θ)i =
(2π)2
Z 2
d k2
×
Bδ (k1 , k2 , −k1 − k2 , w) . (15)(2π)2The term hκ(φ)κ2 (φ + θ)i is a function of θ only. Its contribution
Pµ,2 (s) to the power spectrum of the magnification is given by
the inverse Fourier transform of Eq. (15), which reads
Z 2 0
Z
p3 (w)d s
dw 4κ
Pµ,2 (s) =(2π)2fK (w)!
s −s0 − s
s0
,
,
,w ·
(16)
× Bδ
fk (w) fk (w) fk (w)Fig. 1. The upper panel shows the amplitude of the two first termsof the Taylor expansion of the magnification autocorrelation, namelyhκ(φ)κ(φ + θ)i (dashed line) and 4hκ(φ)κ2 (φ + θ)i (dotted line), using
a source redshift of unity. The sum of these two terms is shown as thesolid line. The lower panel details the relative contribution Rµµ of thesecond-order term for different source redshifts. The figure shows thatthe lowest-order approximation µ ≈ 1 + 2κ misses a substantial part of
the amplitude of the magnification autocorrelation function. Given theaccuracy of the bispectrum fitting formula, Rµµ is accurate to ∼2%.

2.3. Results and predictions
We can now numerically evaluate the first two contributions
to the Taylor expansion of the magnification autocorrelation
function defined in Eq. (5). As mentioned before, we use α = 2
here.
For evaluating the correlation functions, we use a CDM
power spectrum in a spatially flat Universe parameterised with
Ω0 = 0.3, σ8 = 0.9, h = 0.7 and Γ = 0.21. The non-linear evolution of the power spectrum and the bispectrum are computed
according to the formalisms developed by Peacock & Dodds(1996) and Scoccimarro et al. (2000), see Appendix A. The
upper panel of Fig. 1 shows the first- and second-order contributions (dashed and dotted lines, respectively) to the Taylor expansion of the magnification for a fixed source redshift of zs =1. The sum of the two contributions is shown by the solid line.
The figure shows that the contribution of the second-order termreaches an amplitude of more than 30% of the first-order termon angular scales smaller than one arcminute. According to
Eq. (5) which describes the Taylor expansion of the magnification autocorrelation, we define the contribution of the secondorder relative to the first-order term as
Rµµ (θ) =2 α hκ(φ)κ2 (φ + θ)i·hκ(φ)κ(φ + θ)i(17)The lower panel shows this ratio in per cent for different source
redshifts as a function of angular scale. From the lower to the820

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

upper curves, the source redshifts are 1, 1.5, 2 and 3. For eachsource redshift, the contribution of the second term exhibits asimilar dependence on angular scale:– on scales larger than a few degrees, the contribution drops
to negligible values;– effects become relevant on smaller scales, with afairly constant amplitude from a few degrees down toaround 10 arcmin;– on yet smaller scales, the second-order contribution increases steeply, due to the non-linear evolution of the density field. For sources at redshift 2, the amplitude of thesecond term reaches half of the amplitude of the first term
below one arcmin.
Thus, given the amplitude of Rµµ , the correcting term introduced in Eq. (5) is relevant and must be taken into account for
describing the magnification autocorrelation with an accuracy
better than 30%–50% on scales smaller than a few degrees.
So far, we have only investigated the amplitude contributedby the second-order term. In order to estimate the remaining
contributions of all missing terms of the magnification expansion, we will now use numerical simulations allowing a directcomputation of µ as a function of the convergence κ and theshear γ.

3. Magnification statistics from numericalsimulations
On sub-degree scales, lensing effects due to non-linearities
in the density field can only be approximated using analytical fitting formulae (Peacock & Dodds 1996; Scoccimarro &Couchman 2001) as seen above. A full description requires numerical simulations (White & Hu 2000).

3.1. The ray-tracing simulation
For testing the theoretical predictions we performed ray-tracing
experiments in a Very Large N-body Simulation (VLS) recently carried out by the Virgo Consortium (Jenkins et al. 2001;
and see also Yoshida et al. 2001 for simulation details)1 .
The simulation was performed using a parallel P3 M code
(MacFarland et al. 1998) with a force softening length of lsoft ∼30 h−1 kpc. The simulation employed 5123 CDM particles in a
cubic box of 479 h−1 Mpc on a side. It uses a flat cosmologicalmodel with a matter density Ω0 = 0.3, a cosmological constant ΩΛ = 0.7, and a Hubble constant h = 0.7. The initial matter power spectrum was computed using CMBFAST (Seljak& Zaldarriaga 1996) assuming a baryonic matter density ofΩb = 0.04. The particle mass (mpart = 6.86 × 1010 h−1 M ) of
the simulation is sufficiently small to guarantee practically no
discreteness effects on dark-matter clustering on scales down
to the softening length in the redshift range of interest for our
purposes (Hamana et al. 2002).
The multiple-lens plane ray-tracing algorithm weused is detailed in Hamana & Mellier (2001; see also1

The ray-tracing data are available from T. Hamana on request,
hamanatk@cc.nao.ac.jp

Bartelmann & Schneider 1992; Jain et al. 2000 for the theoretical basics); we thus describe only aspects specific to the VLS
N-body data in the following. In order to generate the density
field between z = 0 and z ∼ 3, we use a stack of ten snapshot
outputs from two runs of the N-body simulation, which differ
only in the realisation of the initial fluctuation field. Each cubicbox is divided into 4 sub-boxes of 4792 × 119.75 h−3 Mpc3
with the shorter box side being aligned with the line-of-sightdirection. The N-body particles in each sub-box are projected
onto the plane perpendicular to the shorter box side and thus to
the line-of-sight direction. In this way, the particle distribution
between the observer and z ∼ 3 is projected onto 38 lens planes
separated by 119.75 h−1 Mpc. Note that in order to minimise
the difference in redshift between a lens plane and an output ofN-body data, only one half of the outputs (i.e. two sub-boxes)
at z = 0 are used.
The particle distribution on each plane is converted into thesurface density field on either a 10242 or 20482 regular gridusing the triangular shaped cloud (TSC) assignment scheme(Hockney & Eastwood 1988). The two grid sizes are adopted
for the following reasons:– the 10242 grid is chosen to maintain the resolution provided
by the N-body simulation and removing at the same time
the shot noise due to discreteness in the N-body simulation.
Its computation follows the procedure described in Hamana& Mellier (2001) and Jain et al. (2000). The corresponding
outputs will be labelled with large-scale smoothing in thefollowing.
– the 20482 grid is also chosen to examine effects of smallscale nonlinear structures which are smoothed in the largescale smoothing simulation. We should, however, notethat in this case the shot noise is not sufficiently removed. Actually, the shot-noise power spectrum amplitude
exceeds the convergence power spectrum on scales below ∼1 arcmin. In the following, therefore, we will only
consider measured correlation functions on scales larger
than 1 arcmin. The corresponding outputs will be labelledwith small-scale smoothing below.
Having produced surface density fields on all lens planes,
10242 rays are traced backwards from the observer’s point
using the multiple-lens plane algorithm (e.g. Schneider et al.
1992). The initial ray directions are set on 10242 grids with agrid size of 0.25 arcmin, thus the total area covered by rays
is 4.272 square degrees. We produced 36 realizations of theunderlying density field by randomly shifting the simulation
boxes in the direction perpendicular to the line-of-sight using
the periodic boundary conditions of the N-body boxes. Notethat the lens planes coming from the same box are shifted in
the same way in order to maintain the clustering of matter inthe box.
We point out that second and higher-order statistics ofpoint-source magnifications are generally ill-defined in presence of caustic curves because the differential magnification
probability distribution asymptotically decreases as µ−2 forlarge µ (see Fig. 2). This is a generic feature of magnification near caustics and is thus independent of the lens model.
Strong lensing effects on point sources near caustic curves

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

821

and then identifying two of its three points. In such a case, oneof the correlation lengths of the triple correlator becomes zero,thus necessarily smaller than the smallest relevant scales in any
simulation. This prevents us from using any numerical simulation for directly comparing the results.
In order to avoid this problem, and for comparing our analytical with numerical results, we will introduce an effective smoothing into the theoretical calculations, such that eachvalue of κ at a given position θ is evaluated by averaging theκ-values in a disk of radius θS centred on θ. Indeed, the limit
imposed by the grid size of the simulation gives rise to an unavoidable smoothing-like effect which cancels all information
coming from scales smaller than a corresponding smoothing
scale θS . For this purpose, we introduce a smoothed three-point
correlator,
zsmooth (θ2 − θ1 , θ3 − θ1 ) = hκ(θ1 )κ(θ2 )κ(θ3 )iθS
Z
Z
Z
E
D
=
dθ01 dθ02 dθ03 κ(θ01 )κ(θ02 )κ(θ03 )(18)

× WθS (θ01 − θ1 )WθS (θ02 − θ2 )WθS (θ03 − θ3 ) ,Fig. 2. Probability distribution of the magnification for our small- andlarge-scale smoothing simulations, assuming sources at redshift unity.
The power law tail behaviour (∝µ−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.

give rise to rare, but arbitrarily high magnification values inthe simulations, and therefore the variance of the measured
statistics of µ cannot be defined. However, the smoothing procedure introduced above allows this problem to be removed
because it smoothes out high density regions in the dark matter distribution and thus the fractional area of high magnification decreases. In reality, infinite magnifications do not occur,for two reasons. First, each astrophysical source is extended
and its magnification (given the surface brightness-weighted
point-source magnification across its solid angle) remains finite. Second, even point sources would be magnified by a finite value since for them, the geometrical-optics approximation
fails near critical curves and a wave-optics description leads toa finite magnification (Schneider et al. 1992, Chap. 7).

3.2. Filtering
The computation of correlation functions from numerical simulations is mainly affected by two effects; on large scales by
the finite box size of the dark matter simulation, and on small
scales by the grid size used for computing the surface density
field from the particle distribution. These boundaries set thelimits for the validity of correlation functions measured in numerical simulations. In other words, this means that measuringa correlation function on a given scale is relevant only if thisscale falls within the range of scales defined by the simulation.
As shown in the previous section, our method for computing
the cross-correlation between κ and κ2 consists of first computing a three-point correlation function hκ(φ)κ(φ + θ1 )κ(φ + θ2 )i,

where the function WθS (θ0 ) is a normalised top-hat window ofradius θS . Introducing this smoothing scheme into the expression for hκ(φ)κ2 (φ + θ)i yields
Z
Z 2 Z 2p3 (w)
d s1
d s2
dw 4κ
zsmooth (θS ) =
(2π)2
(2π)2
fK (w)× I(s1 θS ) I(s2 θS ) I(|s1 + s2 |θS )(19)!
s2 −s1 − s2s1,,
, w ei s 1 θ ,× Bδ
fk (w) fk (w) fk (w)where I(x) = 2 J1x(x) . Similarly, introducing the smoothing
scheme into the two-point correlation function gives
wsmooth (|θ2 − θ1 |) = hκ(θ1 )κ(θ2 )iθS!
Z
Zp2κ (w)
s
d2 s
,w
=
dw 2
P
fK (w)(2π)2fK (w)
× |I(s θS )|2 ei sθ .

(20)

The effective smoothing scale depends on two parameters:– the evolution of the apparent grid size of the simulation as
a function of redshift, and– the radial selection function of the dark-matter field whose
correlation function has to be measured.
These quantities are plotted in Fig. 3. In order to use a uniquesmoothing scale valid on the final convergence map, we define
the effective angular smoothing scale byZ
dz W(z) θgrid (z) ,
(21)
θS =
where W(z) is the relevant normalised selection function along
the line-of-sight. Measuring ws means probing the power spectrum along the line-of-sight, weighted by p2κ (z). Therefore, wewill use W(z) = D2+ (z) p2κ (z), where D+ (z) is the growth factor.
In a similar way, we will use W(z) = D4+ (z) p3κ (z) for measuring zsmooth . The numerical values of the corresponding effective
angles are presented in Table 1.

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B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

Fig. 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) = D4+ (z) p3κ (z) (see Eqs. (8) and (21)).
Table 1. Effective smoothing angles in arcminutes for ws and zs computed from Eq. (21) as a function of simulation resolution.

wsmoothzsmoothsmall-scale smoothinglarge-scale smoothingθS = 0.40
θS = 0.39

θS = 0.80
θS = 0.78

The second important difference between analytical calculations and measurements in numerical simulations is the finite
box size effect. Indeed, the analytical correlation functions presented above were computed taking into account all modes in
the power spectrum. However, the finite size of the box used in
the simulation introduces an artificial cutoff in the power spectrum since wavelengths larger than the box size are not sampled
by the simulation. This effect can also be taken into account in
the analytical calculations by simply cancelling all the poweron wavelengths with wave number k < kmin . The boxes we
use have a comoving size of 480 h−1 Mpc which corresponds
to kmin = 0.013 h Mpc−1 .

3.3. Comparing hκ κi and hκ κ2 i
With the help of the filtering schemes introduced in the previous section, we can now compare our theoretical predictions
with correlation functions measured from the numerical simulations. We first compare the amplitude and angular variation of
the two first terms of the Taylor expansion of the magnification
separately. In the next section, we will then compare their sum
to the total magnification fully computed from the simulation.

Fig. 4. Comparison between theoretical predictions and measurementsfrom numerical simulations assuming sources at redshift unity. The
upper and lower curves show hκκi and hκκ2 i, respectively. The points
are measurements from the large-scale smoothing simulations, withthe error bars showing the variance among 36 different realisations.
The dotted lines show the analytical computations taking into accountthe smoothing scale of the simulation. The solid lines additionally include a cut in the power spectrum for cancelling the wavelengths notcovered by the simulation. The dashed line presents the same statistics
without any smoothing. Obviously, the smoothing effects are crucial
for the hκκ2 i cross-correlation.

In Fig. 4, we overplot analytical and numerical results. The
upper curve shows the autocorrelation function of κ as a function of angular scale. We plot in circles the average measurement from 36 realisations of the simulation, and the corresponding 1-σ error bars to show the accuracy of the numericalresults as a function of angular scale. The solid line shows theanalytical prediction, including effective smoothing and an artificial cut of the power at scales below kmin . The agreement isgood on all scales. For comparison, the dotted line shows theresult if we do not impose the large-wavelength cut, and thedashed line is the result if no cut and no smoothing are applied.
In both cases, the deviations from the fully filtered calculation
remain small since we are probing angular scales within therange allowed by the simulation.
The lower curves in Fig. 4 show a quantity proportional to
the second-order correction of the Taylor expansion, namely
the correlation function hκ κ2 i. In the same way as before, thecircles show average measurements from 36 realisations, andthe error bars denote the corresponding 1-σ deviation. The prediction including smoothing and small-wavelength cut (solidline) shows a relatively good agreement given the expected
accuracy of the bispectrum fitting formula, which is approximately 15% (Scoccimarro & Couchman 2000). This time, including smoothing changes the amplitude dramatically, andB. Ménard et al.: Improving the accuracy of cosmic magnification statistics

this effect affects all scales (see the dashed line). As discussed
before, this is expected since we are measuring a three-point
correlator on triangles which have one side length smaller than
the angular grid size of the simulation. Finally, as shown by thedifference between the dotted and solid lines, cancelling thepower on scales where k < kmin again improves the agreement
on large scales.
The agreement between our analytical and numerical computations of hκ κi and hκ κ2 i demonstrates the validity of theformalism introduced in Sect. 2 as well as the choice of theeffective smoothing scale (Eq. (21)) for describing the secondorder term in the Taylor expansion of the magnification.

3.4. Deviations from simulated µ-statistics
We now want to investigate how well the second-order expansion describes the full magnification expression (2) which canbe computed using maps of κ, γ and ω (a net rotation term
which arises from lens-lens coupling and the lensing deflectionof the light ray path; see Van Waerbeke et al. 2001b) obtained
from the simulations (see Hamana et al. 2000 for more detail).
Before doing so, we recall that the amplitude of the magnification autocorrelation measured from the simulation depends
on the smoothing scale, as seen in Sect. 3.2, since µ is nonlinear
in the density field. Therefore, all the following comparisonsare valid for a given effective smoothing length only.
We further emphasise that two problems will complicate
this comparison. First, our analytical treatment is valid in the
weak-lensing regime only, i.e. as long as convergence and shear
are small compared to unity, κ  1, |γ|  1. While mostlight rays traced through the numerical simulations are indeed
weakly lensed, a non-negligible fraction of them will experience magnifications well above two, say. Such events are restricted to small areas with high overdensities and thus affect the magnification statistics only at small angular scales.
Second, a separate problem sets in if and where caustics areformed. The magnification of light rays going through caustics
is infinite, and the magnification probability distribution nearcaustics drops like µ−2 for µ → ∞. As noted above, second- or
higher-order statistics of µ then become meaningless because
they diverge.
Departures of the numerical from the analytical results willthus have two distinct reasons, viz. the occurrence of non-weakmagnifications which causes the analytical to underestimate thenumerical results on small angular scales; and the formation of
caustics, which causes second-order magnification statistics tobreak down entirely. Both effects will be demonstrated below.
They can be controlled or suppressed in numerical simulations
by smoothing, which makes lensing weaker, or by masking
highly magnified light rays or regions containing caustics.
In Fig. 5, we plot with circles the autocorrelation function hµ(φ) µ(φ + θ)i measured from the large- and small-scalesmoothing simulations in the left and right panels, respectively. The presence of caustics is more pronounced in the caseof small-scale smoothing than in the large-scale smoothing
simulations. The dotted line shows the theoretical prediction
given by the first-order term of the Taylor expansion, namely823

4 hκ(φ)κ(φ + θ)i. This yields a low estimate of the correlation,with a discrepancy of order 10% on large scales, and morethan 20% below a few arcminutes.
As expected from the preceding discussion, this level of
discrepancy also depends on the effective smoothing scaleand can increase if simulations with a smaller grid size areused. Estimating the contribution of the two lowest-order termsof µα−1 , we computed in Sect. 2.3 a lower bound to this discrepancy for a real case without smoothing, and found it to reacha level of 25% at large scales, and above 30% below a few arcminutes. The smoothed results taking the additional contribution of the second-order term into account are plotted as solidlines, and give a much better agreement, as expected. To quantify this in more detail, the lower panels of the figure show several contributions compared to the first-order term, i.e. to 4hκκi.
– The symbols show the additional amplitude of the magnification statistics measured from the simulation, comparedto the first-order term also obtained from the simulation,
Rnum
µµ =

hµ(φ)µ(φ + θ)inum−1.
4 hκ(φ)κ(φ + θ)inum(22)

The error bars indicate the 1-σ deviation across 36 realisations.
– The solid line shows the contribution of the second-orderrelative to the first-order term computed from the analyticalexpression including the effective smoothing,=
Rsmoothµµ2αhκ(φ)κ2 (φ + θ)iθShκ(φ)κ(φ + θ)iθS(23)

with α = 2.
In each case, we use the appropriate reference for hκ(φ)κ(φ+θ)i,
i.e. the numerical measurement in the first and the analytical estimation in the second case. Indeed, the measurementof hκ κi from the simulation agrees with the analytical estimation within some uncertainty, which is due to numerical effects
like the finite number of dark-matter boxes used for simulating the light cone. It introduces a bias into our comparisons
which is impossible to separate from the real offset due to all
higher-order terms of the Taylor expansion that were not taken
into account. The two contributions plotted in Fig. 5 are thus ofdifferent nature, but are suitable for a relative comparison.
As the lower panel of the large-scale smoothing simulation shows, the simple 4 hκ κi estimate of the magnification misses 20% of the real amplitude near one arcminute.
This discrepancy almost vanishes after adding the contribution of the second-order term, which gives at all scales a final agreement on the per cent level: the additional amplitude
reaches 19% at the smallest scales of the figure, compared toa value of 20% given by the simulation, and agrees within
better than one per cent on larger scales. Therefore, taking
into account the 2 α hκκ2 i correction allows the accuracy to
be increased by a factor of ∼20 compared to the approximation 4hκκi, in the case of our large-scale smoothing simulation.
On the largest scales, between 6 and 30 arcmin, the agreement
even improves. Above these scales, the numerical results do notallow any relevant comparison because the number of available

824

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

Fig. 5. The upper panel shows a comparison between the magnification autocorrelation measured from the simulation compared to the analyticalestimation for a source redshift of unity. The circles show averaged measurements from 36 realisations of the simulation and the corresponding
1-σ error bars. The dashed line shows the analytical estimation using the approximation µ ≈ 1+ 2κ. The solid line shows the improvement givenby the second-order term of the Taylor expansion of µ. The lower panel shows the relative contribution Rµµ , both measured from the simulationand estimated using expansion terms of µ 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).

independent samplings corresponding to a given separation decreases. On scales below a few arcminutes, the offset between
the measured points and the analytical estimate gives the amplitude of all higher-order terms neglected in the Taylor expansionof the magnification. As we can see, their contribution is on the
one per cent level for the large-scale smoothing simulation.
The curves shown in the right panel demonstrate how theuse of a smaller smoothing scale increases the discrepancy between the analytical and the numerical results. The fraction of
non-weakly magnified light rays increases, and caustics appear
which give rise to a power-law tail in the magnification probability distribution. We investigate the impact of the rare highly
magnified light rays by masking pixels where the simulated
magnification exceeds 4 or 8, and show that caustics have no
noticeable effect on the amplitude of the magnification autocorrelation function determined from these simulated data. Note,however, that the impact of the caustics depends on the source
redshift. The higher the redshift, the more caustics appear, andthe larger is their impact on the correlation amplitude.
Imposing lower masking thresholds removes a significant
fraction of the area covered by the simulation, changing the
spatial magnification pattern and thus the magnification autocorrelation function. The corresponding measurements are
represented by the dashed error bars in the lower right panelof Fig. 5. We note that the error bars of Rnumµµ computed withthe small-scale smoothing simulation become larger at small
scales compared to the lower left panel. This reflects the factthat second-order magnification statistics are ill-defined once
caustics appear. In the next section, we will investigate similarsmoothing effects on cross-correlations between magnificationand dark matter fluctuations. These quantities are not affectedby problems of poor definition when the smoothing scale becomes small, and therefore do not show larger error bars at
small scales when the smoothing scale decreases.
These comparisons show that the approximation µ ≈ 1 + 2κmisses a non-negligible part of the total amplitude of weaklensing magnification statistics. The formalism introduced in
Sect. 2 allows second-order corrections to be described with orwithout smoothing of the density field. This provides a better
description of the correlation functions, but still gives a lower
amplitude than the simulation results. As we noticed, the analytic computation based on the Taylor expansion is sufficiently
accurate only in the weak lensing regime. In reality, however,the strong lensing, which can not be taken into account in theanalytic formalism, has a significant impact on the magnification correlation especially at small scales as shown in the smallscale smoothing simulation. Therefore, one should carefully
take the strong lensing effect into consideration when one interprets the magnification related correlation functions. However,
we will see in the next section that counts-in-cells estimators
are less affected by the strong lensing than correlation functions and thus enable better comparisons of observations withresults from simulations.

4. Applications to quasar-galaxy correlations
As a direct application of the formalism introduced previously, we now investigate the effects of second-order terms
on a well-known magnification-induced correlation, namely

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

825

the quasar-galaxy cross-correlation (the results can also be
applied to galaxy-galaxy correlations induced by magnification; Moessner & Jain 1998). In order to estimate cosmologicalparameters from this kind of correlations, we then suggest theuse of a more suitable estimator using counts-in-cells rather
than two-point correlation functions. It has the advantage ofmaking the observational results more easily reconciled withthe ones from numerical simulations.

4.1. Formalism and correcting terms
The magnification bias of large-scale structures, combinedwith galaxy biasing, leads to a cross-correlation of distant quasars with foreground galaxies. The existence of thiscross-correlation has firmly been established (e.g. Benı́tez &Martı́nez-González 1995; Williams & Irwin 1998; Norman &Impey 1999; Norman & Williams 2000; Benı́tez et al. 2001;
Norman & Impey 2001). Ménard & Bartelmann (2002) showedthat the Sloan Digital Sky Survey (York et al. 2000) will allow
this correlation function to be measured with a high accuracy.
Its amplitude and angular shape contain information on cosmological parameters and the galaxy bias factor. Thus, it is important to accurately describe these magnification-related statistics
in order to avoid a biased estimation of cosmological parameters as well as the amplitude of the galaxy bias.
As shown in Bartelmann (1995), the lensing-induced crosscorrelation function between quasars and galaxies can be
written as
wQG (θ) ≡ hδQSO (φ) δgal (φ + θ)i= hδµα−1 (φ) δgal (φ + θ)i .

(24)

Using the above formalism, we can expand the effective magnification fluctuation δµα−1 up to second order and find the correcting term:hi(25)wQG (θ) = 2 (α − 1) hκδgal i + α hκ2 δgal i .
The second term is proportional to α (contrary to the factor 2α
in Eq. (5)), since there is only one contribution of the magnification. Therefore, the expected effects will be roughly halfof those on the autocorrelation of the effective magnification
seen in the previous section. Assuming a linear bias b between
galaxies and dark matter, the cross-correlation between δgaland κ2 can be written ashδgal (φ)κ2 (φ + θ)i = b hδDM (φ) κ2 (φ + θ)i
Z 2 Z 2Z
p2 (w) pδ (w)
d s1
d s2
= dw κ 4
(2π)2
(2π)2
fK (w)!
s2 −s1 − s2s1,,
, w eis1 θ , (26)× Bδfk (w) fk (w) fk (w)where pδ (w) is the normalised distance distribution of thegalaxies. For this example, we will use

!β 

β z2
z 
exp −pδ (z) dz = 3(27) dz ,z0z0 Γ(3/β)with β = 1.5 and z0 = 0.3.

Fig. 6. The upper panel shows the amplitude of the normalised quasargalaxy correlation wQG /2(α − 1) as a function of angular separation.
We show the first two terms of the Taylor expansion of this correlation,
namely hδ(φ)κ(φ + θ)i (dashed line) and 2hδ(φ)κ2 (φ + θ)i (dotted line),using a source redshift of unity. The sum of these two terms is shownas the solid line. The lower panel details the relative contribution Rδµand of the second-order term for different source redshifts, namely
z = 1, 1.5, 2 and 3 from bottom to top.

The results are shown in Fig. 6. As we can see, previous
estimates using the approximation µ ≈ 2 κ missed approximately 15% of the amplitude on small scales for quasars at
redshift unity. Using quasars at redshift 2, these effects reachup to 25%. These offsets, which are only lower limits, would
lead to biased estimates of Ω0 or b, for example.
As for the magnification autocorrelation, we can compare
our theoretical estimates against numerical estimations. We canfirst introduce a coefficient Rδµ describing the accuracy of oursecond-order correction:Rδµ (θ) =α hδ(φ)κ2 (φ + θ)i·hδ(φ)κ(φ + θ)i(28)

We plot the results in Fig. 7. Note that contrary to the magnification autocorrelation, this quantity does not suffer from poor
definition, even without smoothing. The difference can be seenby the same size of the error bars between the two simulation
results at small scales, whereas they were larger in the caseof hµµi for the small-scale smoothing simulation (Fig. 5). The
results for Rδµ are very similar those obtained for Rµµ : for thelarge-scale smoothing ray-tracing we find very good agreement
which reaches the one percent level on small scales. However,when the smoothing length decreases, we see from the smallscale smoothing outputs that we are missing a part of the total
amplitude on small scales, which shows that higher-order terms
play a non negligible role on those scales.

826

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics

arcminutes will not be relevant. Using galaxies as backgroundsources, this limitation occurs at much smaller scales.
Using a first-order Taylor expansion for the magnification,
the new estimator w̄QG (θ) can be writtenZ
Zw̄QG (θ)
pκ (w)pδ (w)sds=dw
2 (α − 1) b
2π
fK2 (w)!
s, w |I(s θ)|2 ,(30)
×P
fk (w)

Fig. 7. Comparison of Rδµ given by the theoretical calculation andmeasured from the numerical simulation. The results are shown forthe large- and small-scale smoothing simulations from bottom totop. They show that a second-order description of the hµδi crosscorrelation 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.

4.2. Using count-in-cells estimators
For precisely estimating cosmological parameters as well as
the amplitude of the galaxy bias, it is necessary to employ theoretical magnification statistics that closely describe the observables. However, we have seen in Sect. 3 that analyticalestimates as well as numerical simulations have intrinsic limitations and prevent us from accurately describing usual n-point
correlation functions related to magnification statistics.
Besides, it is possible to focus on another estimator closely
related to correlation functions, namely a count-in-cells estimator, which naturally smoothes effects originating from thedensity field and can thus more easily be reconciled with numerical simulations. So far, quasar-galaxy or galaxy-galaxy
correlations have been quantified measuring the excess ofbackground-foreground pairs at a given angular separation.
Instead, we can correlate the amplitude of the background andforeground fluctuations, both measured inside a given aperture.
We will therefore introduce a count-in-cells estimator,
D
Ew̄QG (θ) = δQSO (φ) δgal (φ)
Eθ
D
α−1
(29)
δµ (φ) δgal (φ) ,θwhere the subscript θ indicates averaging of δQSO (φ) and δgal (φ)
inside a cell of radius θ. In practice, this estimator is intended
to be applied to galaxy-galaxy rather than to quasar-galaxy correlations, since the average angular separation between bright
distant quasars is of order one degree for current surveys, thus
averaging the source counts inside cells with radii of several

where I(x) = 2 J1x(x) . This expression differs from the 2-point
correlation function (9) by its Fourier-space filtering of thepower spectrum. The additional smoothing wipes out the poweron scales smaller than the physical scale corresponding to the
angular smoothing scale θ. For any observational result to becompared to a numerical simulation, θ and the smoothing scale
used in the simulation will have to be carefully adapted to eachother and to the redshift distribution of the foreground galaxy
distribution.
In practice, masking always makes correlation functions
easier to measure than counts-in-cells. However, in a large
survey with short exposures like the SDSS, masking is not areal issue to measure counts-in-cells since unusable regions arequite rare and their area is small compared to the total survey
size. This is different for cosmic shear surveys for which images are deeper and saturation occurs more frequently.
Note that gravitational lensing by the foreground galaxies themselves is entirely irrelevant here. The angular scale onwhich galaxies act as efficient lenses is on the order of one
arc second and below, much smaller than the angular scales
we are concerned with. Moreover, the probability for a quasar
to be strongly lensed by a galaxy is well below one per cent.
Bartelmann & Schneider (1991) demonstrated this point explicitly by including galaxies into their numerical simulations
and showing they had no noticeable effect.

5. Conclusion
As surveys mapping the large-scale structure of the Universe
become wider and deeper, measuring cosmological parameters
as well as the galaxy bias with cosmic magnification will become increasingly efficient and reliable. Therefore, an accurate
theoretical quantification of magnification statistics becomesincreasingly important.
Previous estimates of cosmic magnification relied on theassumption that the magnification deviates sufficiently little
from unity that it can be accurately approximated by its firstorder Taylor expansion about unity, i.e. µ ≈ 1 + 2 κ. In thispaper, we have tested the validity of this assumption in theframework of magnification statistics, by investigating thesecond-order terms in the Taylor expansion of µ. We haveshown that:– Second-order terms can be related to the cross-correlation
between κ and κ2 ;– their importance increases as the number-count function of
the background sources steepens, i.e. as α increases;

B. Ménard et al.: Improving the accuracy of cosmic magnification statistics– their amplitude is not negligible: for the magnification autocorrelation, their contribution is typically on the order of
30%–50% at scales below one degree. Therefore, previous
estimates of cosmic magnification were systematically low.
For testing our theoretical calculations, we have compared our
results to magnification statistics found in numerical simulations by performing ray-tracing experiments in a very large
N-body simulation. We have first checked the validity of ourformalism describing the correlation hκκ2 i, and demonstrated
the importance of including an effective smoothing into the analytical calculations. Indeed, µ is nonlinear in the density field
and the amplitude of magnification statistics measured fromnumerical simulations depends therefore on the available resolution.
Using a simulation with an effective smoothing scaleof 0.8 arcmin, we found that our second-order formalism isaccurate to the percent level for describing magnification autocorrelations. Compared to previous estimates, this improves
the accuracy by a factor of ≈20. For smaller effective smoothing scales, the contribution of third- and higher-order terms becomes important on scales below a few arcminutes.
Finally we have applied our formalism to observed correlations, like quasar-galaxy and galaxy-galaxy correlations
due to lensing. We have shown that second-order corrections
increase their amplitude by 15% to 25% on scales below one
degree. These correlations are valuable tools to probe cosmological parameters as well as the galaxy bias. However,even including our correcting terms, analytical or numericalestimates of magnification statistics can only provide lower
bounds to the real amplitude of the correlation functions in the
weak-lensing regime. Thus, we propose using count-in-cells
estimators rather than correlation functions since the intrinsic
smoothing in determining counts-in-cells allows the observational results to be more directly related to those obtained innumerical simulations.
Thus, some care is required in using cosmic magnification
as described by a Taylor expansion for constraining cosmological parameters, especially for interpreting measurements
on small angular scales. Therefore, describing magnification
statistics using the halo-model formalism will be of great interest in order to achieve a precise and direct description ofobservational quantities.
Acknowledgements. We thank Francis Bernardeau and Stéphane
Colombi for helpful discussions. This work was supported in partby the TMR Network “Gravitational Lensing: New Constraints on
Cosmology and the Distribution of Dark Matter” of the EC under contract No. ERBFMRX-CT97-0172.

827

where δ(2) is of order (δ(1) )2 and represents departures fromGaussian behaviour, yields the bispectrum(1) (1)(1) (1) (2)hδ1 δ2 δ3 i ' hδ(1)1 δ2 δ3 i + hδ1 δ2 δ3 i
+ cyclic terms (231, 312) .

(A.2)

The first term in Eq. (A.2) vanishes because the density fluctuation field is Gaussian to first order, hence the third momentof δ(1) is zero. Thus, the leading term in Eq. (A.2) is of the(1) (2)order of hδ(1)1 δ2 δ3 i and can be quantified using second-order
perturbation theory.
The bispectrum Bδ (k1 , k2 , k3 ) is defined only for closed
triangles formed by the wave vectors k1 , k2 , k3 . It can be expressed as a function of the second-order kernel F(k1 , k2 ) andthe power spectrum
Bδ (k1 , k2 , k3 ) = 2 F(k1 , k2 ) P(k1 )P(k2 )+2 F(k2 , k3 ) P(k2 )P(k3 )+2 F(k1 , k3 ) P(k1 )P(k3 ) .

(A.3)

For describing the bispectrum on all scales, we use the fitting
formula derived by Scoccimarro & Couchman (2000) for the
non-linear evolution of the bispectrum in numerical simulations of CDM models, extending previous work for scale-free
initial conditions. In that case, we haveF(k1 , k2 ) =5a(n, k1 )a(n, k2)
7!
1 k 1 · k 2 k1 k2++
b(n, k1 )b(n, k2)
2 k1 k2 k2 k1!2
2 k1 · k2
c(n, k1)c(n, k2 ),+7 k1 k2

(A.4)

with
a(n, k) =b(n, k) =c(n, k) =1/2(q/4)n+3.5
1 + σ−0.2
8 (z) [0.7 Q3 (n)]1 + (q/4)n+3.5
1 + 0.4 (n + 3) qn+3
1 + qn+3.5
hi
1 + 4.5/ 1.5 + (n + 3)4 (2q)n+3
1 + (2q)n+3.5

,

(A.5)

3
PL (kNL ) = 1, and PL (k) is
and q ≡ k/kNL (z), where 4πkNL
the linear power spectrum at the desired redshift. The effective spectral index is taken from the linear power spectrum as
well. The function Q3 (n) is given by

Q3 (n) =(4 − 2n )·(1 + 2n+1 )(A.6)

For more detail, see Scoccimarro & Couchman (2000).

Appendix A: Bispectrum and non-linear evolution
The bispectrum can be estimated using second-order perturbation theory. Indeed, an expansion of the density field to second
nonlinear order asδ(x) ≈ δ(1) (x) + δ(2) (x) ,(A.1)

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