Mon. Not. R. Astron. Soc. 346, 949–962 (2003)

Halo model predictions of the cosmic magnification statistics: the full
non-linear contribution
Masahiro Takada1  and Takashi Hamana2,3 
1 Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104, USA
2 Institut d’Astrophysique de Paris, 98 bis Bld Arago, F-75014, Paris, France
3 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan

Accepted 2003 August 23. Received 2003 August 13; in original form 2003 May 20

ABSTRACTThe lensing magnification effect due to large-scale structure is statistically measurable by thecorrelation of size fluctuations in distant galaxy images as well as by cross-correlation between
foreground galaxies and background sources such as the QSO–galaxy cross-correlation. We
use the halo model formulation of Takada & Jain to compute these magnification-induced
correlations without employing the weak lensing approximation, µ ≈ 1 + 2κ. Our predictions
thus include the full contribution from non-linear magnification, δµ  1, that is due to lensing
haloes. We compare the model prediction with ray-tracing simulations and find excellent
agreement over the range of angular scales we consider (0.5  θ  30 arcmin). In addition,
we derive the dependence of the correlation amplitude on the maximum magnification cutoff µmax , which it is necessary to introduce in order to avoid the contributions from strong
lensing events. For a general correlation function parametrized as µp f  (f is any cosmic fieldcorrelated with the magnification field), the amplitude remains finite for p < 1 and diverges
for p  1 as µmax → ∞, independent of the details of the lensing mass distribution and of theseparation angle. This consequence is verified by the halo model as well as by the simulations.
Thus, the magnification correlation with p  1 has a practical advantage in that it is insensitive
to a selection effect of how strong lensing events with µ  1 are observationally excluded
from the sample.
The non-linear magnification contribution enhances the amplitude of the magnification correlation relative to the weak lensing approximation, and the non-linear correction is moresignificant on smaller angular scales and for sources at higher redshifts. The enhancement
amounts to 10–25 per cent on arcmin scales for the QSO–galaxy cross-correlation, even after
the inclusion of a realistic model of galaxy clustering within the host halo. Therefore, it isnecessary to account for the non-linear contribution in theoretical models in order to make anunbiased, cosmological interpretation of the precise measurements expected from forthcoming
massive surveys.
Key words: gravitational lensing – cosmology: theory – large-scale structure of Universe.

1 INTRODUCTION
Gravitational lensing caused by the large-scale structure is now recognized as a powerful cosmological tool; see Mellier (1999) and
Bartelmann & Schneider (2001) for thorough reviews. The gravitational deflection of light causes an increase or decrease of the area ofa given patch on the sky depending on whether the light ray passes
preferentially through the overdense or underdense region. Accordingly, this causes an observed image of source to be magnified or demagnified relative to the unlensed image, because lensing conserves
 E-mail: mtakada@hep.upenn.edu (MT); hamana@iap.fr (TH)C

2003 RAS

the surface brightness and the received luminosity is proportional to
the solid angle of the image. Large magnifications are observed in astrong lensing system that accompanies multiple images or largely
deformed images. It has also been proposed that mild or weak magnification is measurable in a statistical sense. The magnification leads
to an enhancement in the flux-limited number counts of backgroundsources around the foreground sample that traces the lensing mass
distribution. Based on this idea, numerous works have investigatedthe QSO–galaxy cross-correlation theoretically (e.g. Broadhurst,
Taylor & Peacock 1995; Bartelmann 1995) as well as observationally (e.g. Benı́tez & Martı́nez-González 1997; Benı́tez, Sanz &Martı́nez-González 2001; Gaztañaga 2003; see also Bartelmann &Schneider 2001 for a thorough review). Further, Jain (2002)

950

M. Takada and T. Hamana

recently proposed that the magnification effect can be extracted
by statistically dealing with size fluctuations of distant galaxy images, although this is a great challenge for existing data as yet.
Forthcoming massive surveys such as the Sloan Digital Sky Survey (SDSS),1 the Deep Lens Survey (DLS),2 the Canada–France–
Hawaii Telescope (CFHT) Legacy Survey3 as well as future precise imaging surveys such as the Supernova/Acceleration Probe(SNAP),4 the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS)5 and the Large-aperture Synoptic Survey Telescope (LSST)6 allow measurements of these magnification effects athigh significance. Therefore, it is of great importance to explore how
this type of method can be a useful cosmological tool and complementary to the established cosmic shear method, which measurescorrelation between lensing-induced ellipticities of distant galaxy
images (e.g. Hamana et al. 2003; Jarvis et al. 2003, and references
therein).
The magnification field µ(θ) in a given direction θ on the sky isexpressed (e.g. Schneider, Ehlers & Falco 1992) asµ(θ) = |[1 − κ(θ)]2 − γ 2 (θ)|−1 .

(1)Here κ and γ are the convergence and shear fields, which are fully
determined by the mass distribution along the line of sight. Thisequation shows the non-linear relation between the magnificationand the convergence and shear, and indicates that the magnification
increases with κ and γ very rapidly and becomes even formallyinfinite when the lensing fields κ, γ ∼ O(1). However, as long as
we are concerned with the magnification related statistics due tothe large-scale structure, strong lensing events (µ  1) should be
removed from the sample to prevent the large statistical scatters.
This will be straightforward to implement, if the strong lensing
accompanies multiple images or largely deformed images. On theother hand, modest magnification events (δµ  1) make it relatively
difficult to identify and are likely included in the sample for theblind analysis, because the magnification is not a direct observable.
Therefore, the magnification statistics rather requires a more careful
study of the selection effect than the cosmic shear (e.g. Barber &Taylor 2003), which we will carefully address.
The simplest statistical quantity most widely used in cosmology
is the two-point correlation function (2PCF). For our purpose, themagnification field is taken as either or both of the two fields entering
into the correlation function. However, it is not straightforward toanalytically compute the magnification 2PCF because of the nonlinear relation between µ and κ, γ , where the latter fields are easier
to compute in a statistical sense based on a model of the mass
power spectrum. For this reason, the conventional method of themagnification statistics employs the weak lensing approximationµ ≈ 1+2κ (e.g. Bartelmann 1995; Dolag & Bartelmann 1997; Sanz,
Martinez-González & Benitez 1997; Benı́tez & Martı́nez-González
1997; Moessner & Jain 1998; Benı́tez, Sanz & Martı́nez-González
2001; Ménard & Bartelmann 2002; Gaztañaga 2003; Jain, Scranton& Sheth 2003). However, it is obvious that this type of method isvalid only in the limit κ, γ  1 and likely degrades the model
accuracy on non-linear small scales. In fact, using the ray-tracing
simulations Ménard et al. (2003) clarified the importance of the
non-linear magnification contribution to the magnification statistics
1 http://www.sdss.org/
2 http://dls.bell-labs.com/
3 http://www.cfht.hawaii.edu/Science/CFHLS/
4 http://snap.lbl.gov
5 http://www.ifa.hawaii.edu/pan-starrs/
6 http://www.dmtelescope.org/dark home.html(see also Barber & Taylor 2003). It was shown that the perturbative
treatment breaks down over a range of angular scales of our interest.
Although a promising method to resolve this issue is to employ
ray-tracing simulations, to perform multiple evaluations in model
parameter space requires a sufficient number of simulation runs,
which is relatively prohibitive.
Therefore, the main purpose of this paper is to develop an analytical method to compute the magnification-induced correlation
function without employing the weak lensing approximation. To dothis, we use the halo model to describe gravitational clustering in
the large-scale structure, following the method developed in Takada& Jain (2003a,b,c, hereafter TJ03a,b,c). The model prediction of
the QSO–galaxy cross-correlation is also developed by incorporating a realistic model of galaxy clustering within the host halo
into the halo model. Although the halo model rather relies on thesimplified assumptions, the encouraging results revealed so far arethat it has led to consistent predictions to interpret observational results of galaxy clustering as well as to reproduce simulation results(see, for example, Seljak 2000; Zehavi et al. 2003; Takada & Jain
2002, hereafter TJ02; TJ03a,b,c; also see Cooray & Sheth 2002 for areview).
Another purpose of this paper is to explore how the magnification
related statistics can probe the halo structure. The non-linear magnifications arise when the light ray emitted from a source encounters
an intervening mass concentration, i.e. a dark matter halo such asa galaxy or cluster of galaxies. It is known that strong lensing ofµ  1 can be used to probe detailed mass distribution within ahalo (e.g. Hattori, Kneib & Makino 1999). Similarly, modest nonlinear magnifications of δµ  1 could lead to a sensitivity of themagnification statistics to the halo structure in a statistical sense, as
investigated in this paper. A fundamental result of cold dark matter(CDM) model simulations is that the density profiles of haloes areuniversal across a wide range of mass scales (e.g. Navarro, Frenk &White 1997, hereafter NFW). On the other hand, some alternativemodels such as the self-interacting dark matter scenario (Spergel& Steinhardt 2000) have been proposed in order to reconcile thepossible conflicts between the simulation prediction and the observation. If dark matter particles have a non-negligible self-interaction
between themselves, the effect is likely to yield a drastic change on
the halo profile compared to the CDM model prediction (Yoshida
et al. 2000). The halo structure thus reflects the dark matter nature
as well as the detailed history of non-linear gravitational clustering.
Hence, exploring the halo profile properties with gravitational lensing can be a direct test of the CDM paradigm on scales Mpc,
which is not attainable with the cosmic microwave backgroundmeasurement.
The plan of this paper is as follows. In Section 2 we present
the basic equations relevant for cosmological gravitational lensing
and then briefly summarize two promising methods to statisticallymeasure the magnification effect. In Section 3 we develop an analytical method to compute the magnification correlation functions
based on the halo model. In Section 4, we derive an asymptotic
behaviour of the correlation amplitude for large magnifications. In
Section 5 we qualitatively test the halo model prediction and theasymptotic behaviour using ray-tracing simulations. The realistic
model of the QSO–galaxy cross-correlations is also derived. Section 6 is devoted to a summary and discussion. Throughout this paper, without being explicitly stated, we consider the CDM modelwith m0 = 0.3, λ0 = 0.7, b0 , h = 0.7 and σ8 = 0.9. Herem0 , b0 and
λ0 are the present-day density parameters of matter,baryons and the cosmological constant, h is the Hubble parameter,and σ 8 is the rms mass fluctuation in a sphere of radius 8 h−1 Mpc.
C

2003 RAS, MNRAS 346, 949–962

Magnification statistics
Fourier space, we have the simple relation between κ and γ

2 PRELIMINARIESγ̃1 (l) = cos 2ϕl κ̃(l),2.1 Magnification of gravitational lensδθiS = Ai j δθ JI ,(2)

where δθ i is the separation vector between points on the respective planes. The lensing distortion of an image is described by the
Jacobian matrix Ai j defined asA=

1 − κ − γ1

−γ2

−γ2

1 − κ + γ1



,

(3)

where κ is the lensing convergence and γ1 and γ2 denote the tidal
shear fields, which correspond to elongation or compression alongor at 45◦ to the x-axis, respectively, in the given Cartesian coordinate
on the sky. κandγi depend on angular position, although we haveomitted showing this in the argument for simplicity. Because thegravitational lensing conserves surface brightness from a source,
the lensing magnification, the ratio of the flux observed from theimage to that from the unlensed source, is given by determinationof the deformation matrix, yielding equation (1). In the weak lensing limit κ, γ  1, we can Taylor expand the magnification field
as µ ≈ 1 + 2κ, as conventionally employed in the literature tocompute the magnification statistics. The weak lensing approximation ceases to be accurate as κ, γ → O(1). For example, κ = 0.5(and simply γ = 0) leads to a factor of 2 difference as µ = 4 and1 + 2κ = 2.
In the context of cosmological gravitational lensing, the convergence field is expressed as a weighted projection of the threedimensional (3D) density fluctuation field between source andobserver χHκ(θ) =dχ W (χ )δ[χ , dA (χ )θ],(4)0

where χ is the comoving distance, dA (χ ) is the comoving angular diameter distance to χ, and χ H is the distance to the Hubble horizon. Note that χ is related to redshift z via the relationdχ = dz/H(z) (H(z) is the Hubble parameter at epoch z). Following
the pioneering work carried out by Blandford et al. (1991), MiraldaEscude (1991) and Kaiser (1992), we used the Born approximation,
where the convergence field is computed along the unperturbed path,neglecting higher-order terms that arise from coupling between two
or more lenses at different redshifts. Using ray-tracing simulationsof the lensing fields, Jain, Seljak & White (2000) proved that the
Born approximation is a good approximation for lensing statistics(see also Van Waerbeke et al. 2001; Vale & White 2003). The lensing
weight function W is given by
W (χ ) =3
2

2 −1
m0 H0 a (χ ) dA (χ )

 χH

×

dχs f s (χs )χdA (χs − χ ),dA (χs )

(5)

where
 χH fs (χs ) is the redshift selection function normalized asdχ f s (χ ) = 1. In this paper we assume all sources are at a0single redshift zs for simplicity; fs (χ ) = δD (χ − χs ). H0 is the
Hubble constant (H0 = 100 h km s−1 Mpc−1 ). Similarly, the shear
fields are derivable from the density fluctuation fields, but the relation is non-local due to the nature of the gravitational tidal force. In
C

2003 RAS, MNRAS 346, 949–962γ̃2 (l) = sin 2ϕl κ̃(l),(6)

where the Fourier-mode vector is l = l(cos ϕl , sin ϕl ).

The gravitational deflection of a light ray induces a mapping between
the two-dimensional source plane (S) and the image plane (I) (e.g.
Schneider et al. 1992)



951

2.2 Methodology for measuring the magnification statistics
There are two promising ways for measuring the lensing magnification effect statistically, which are likely feasible for forthcoming
and future surveys. Here we briefly summarize the methodology.
Gravitational magnification has two effects. First, it causes thearea of a given patch on the sky to increase, thus tending to dilute thenumber density observed. Secondly, sources fainter than the limiting
magnitude are brightened and may be included in the sample. The net
effect, known as the magnification bias, depends on how the loss ofsources due to dilution is balanced by the gain of sources due to flux
magnification. Therefore, it depends on the slope s of the unlensed
number counts of sources N0 (m) in a sample with limiting magnitudem, s = d log N0 (m)/dm. Magnification by amount µ changes thenumber counts to (e.g. Broadhurst et al. 1995; Bartelmann 1995)
N  (m) = N0 (m)µ2.5s−1 .

(7)

For the critical value s = 0.4, magnification does not change thenumber density; it leads to an excess for s > 0.4, and a deficit fors < 0.4. Let n1 (θ) be the number density of the foreground sample
with mean redshift z1 , observed in the direction θ on the sky, andlet n2 (θ) be that of the source sample with a higher mean redshiftz2 > z1 . Thus, even if there is no intrinsic correlation between
the two populations, magnification induces the non-vanishing crosscorrelation
ξ (θ) = δn 1 (θ 1 )δn 2 (θ 2 )|θ1 −θ2 |=θ= δn 1 (θ 1 )[µ(θ 2 )]2.5s−1 |θ −θ |=θ ,
1

(8)

2

where δn i (θ) = (n i (θ) − n̄ i ))/n̄ i , with n̄ i being the average number
density of the ith sample. Here · · · denotes the ensemble average and observationally means the average over all pairs separatedby θ on the sky. Based on this idea, numerous studies have confirmed the existence of the enhancement of the QSO number counts
around foreground galaxies, i.e. the QSO–galaxy cross-correlation(see, for example, Benı́tez et al. 2001 and Gaztañaga 2003, andreferences therein; also see Bartelmann & Schneider 2001 for a review). Although these results are in qualitative agreement with themagnification bias, in most cases the amplitude of the correlation ismuch higher than that expected from gravitational lensing models.
The excess might be due to the non-linear magnification contribution from massive haloes (see for example, the discussion aroundfig. 28 in Bartelmann & Schneider 2001). However, to make such astatement with confidence, it is necessary to further explore an obstacle in the theoretical model, the bias relation between the galaxy
and mass distributions. This still remains uncertain observationallyand theoretically. In particular, on small scales (Mpc), it is crucial to model how to populate haloes of given mass with galaxies,
known as the halo occupation number (e.g. Seljak 2000; TJ03b).
Recently, Jain et al. (2003) carefully examined the effect of the halo
occupation number on the magnification bias and showed that theparameters used in it yield a strong sensitivity to the predicted correlation. For example, possible modification for types of galaxies
leads to a change of a factor of 2–10 in the expected signal on arcmin scales. Further, we will show that the non-linear magnification
correction is also important to make the accurate prediction.
Secondly, Jain (2002) recently proposed a new method for measuring the magnification effect of the large-scale structure, based

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M. Takada and T. Hamana

on the work of Bartelmann & Narayan (1995). The lensing effectis to increase or decrease an observed image of a galaxy relative
to the unlensed image, depending on whether the light ray travels preferentially through the overdense or underdense region that
corresponds to µ > 1 or µ < 1. That is, the area S and characteristic radius R(∝ S1/2 ) are changed by the magnification fieldµ as
S → Sµ,R → Rµ1/2 .

(9)

Although the unlensed image is not observable, this effect can bestatistically extracted as follows. We can first estimate the mean sizeof source galaxies from the average over the sample available froma given survey area, under the assumption µ = 1 or µ1/2  = 1.
Then, the 2PCF of the size fluctuations can be computed from theaverage over all pairs of galaxies separated by the angle considered,in analogy with the 2PCF of the cosmic shear fields (the variancemethod was considered in Jain 2002). The reason that this method is
not yet feasible is that it requires a well-controlled estimate of the unlensed size distribution as well as systematics (photometric calibration, resolution for sizes and PSF anisotropy). Hence, space-based
imaging surveys will make possible the measurement of galaxy sizes
with a sufficient accuracy hard to achieve from the ground so far.
This method can potentially be a precise cosmological probe as thecosmic shear measurement, because it is free from galaxy bias uncertainty. Further, the non-linear nature of the magnification could
lead to complementarity to the cosmic shear measurement, as we
will discuss below.

3.1 Halo profile and mass function
We use the the halo model of gravitational clustering to computethe magnification statistics, following the method in TJ03a,b,c. Key
model ingredients are the halo profile ρh (r), the mass function n(M)and the halo bias b(M), each of which is well investigated in theliterature.
As for the halo profile, we employ an NFW profile truncatedat radius r180 , which is defined so that the mean density enclosedby sphere with r180 is 180 times the background density. Within aframework of the halo model, we need to express the NFW profile
in terms of M 180 and redshift z. To do this, we first express thetwo parameters of the NFW profile, the central density parameterand the scale radius, in terms of the virial mass and redshift, basedon the spherical top-hat collapse model and the halo concentration
parameter of Bullock et al. (2001) (see TJ03b,c for more details).
Then, following Hu & Kravtsov (2003), we employ a conversion
method between the virial mass and M 180 in order to re-express the
NFW profile in terms of M 180 and z. In what follows, we will often
refer to halo mass M as M 180 for simplicity.
For the mass function, we employ the Sheth–Tormen mass function (Sheth & Tormen 1999)ρ̄0n(M)dM =f (ν) dν
M


√
ρ̄0
aν dν
−p
(10)
A[1 + (aν) ] aν exp −
,
=
M
2
ν
where ν is the peak height defined byδc (z)
D(z)σ (M)2.

3.2 Lensing fields for an NFW profile
For an NFW profile, we can derive analytical expressions to givethe radial profiles of convergence, shear and magnification.
For a given source at redshift zs , the convergence profile for a
halo of mass M at z, denoted by κM , can be given as a function of
the projected radius from the halo centreκ M (θ) = 4πGa −1

dA (χ)dA (χs − χ) M f
F(θ/θs ),dA (χs )
2πrs2

(12)

where rs is the scale radius, θs is its projected angular scale andf = 1/[ln(1 + c180 ) − c180 /(1 + c180 )] (see below for the definitionof c180 ). The explicit form of F(x) is given by equation (27) in TJ03b.
For an axially symmetric profile, the shear amplitude can be derivedas

3 H A L O A P P R OAC H T O M AG N I F I C AT I O NS TAT I S T I C Sν=Here σ (M) is the present-day rms fluctuation in the mass density,smoothed with a top-hat filter of radius R M ≡ (3M/4πρ̄0 )1/3 , δc isthe threshold overdensity for the spherical collapse model and D(z)is the linear growth factor (e.g. Peebles 1980). The numerical coefficients a and p are taken from the results of table 2 in White (2002) as
a = 0.67 and p = 0.3, which are different from the original values
a = 0.707 and p = 0.3 in Sheth & Tormen (1999). Note that thenormalization coefficient A = 0.129. The main reason we employ
the halo boundary r180 is that the mass function measured in N-bodysimulations can be better fitted by the universal form (10) when we
employ the halo mass estimator of M 180 , rather than using the virial
mass estimator, as carefully examined in White (2002). To maintainconsistency, we also employ the halo biasing b(M) given in Sheth &Tormen (1999) with the same a and p parameters,
which is needed forthe two-halo term calculation. Note that dMb(M)n(M)M/ρ̄0  1
for the halo bias model and the mass function we employ (e.g. Seljak
2000).

(11)γ M (θ) = κ̄(θ) − κ(θ)
dA (χ)dA (χs − χ) M f= 4πGa −1
G(θ/θs ),dA (χs )
2πrs2

(13)

where G(x) is given by equation (16) in TJ03c. These expressionsof κM and γM differ from those given in Bartelmann (1996), which
are derived from an infinite line-of-size projection of the NFW profile under the implicit assumption that the profile is valid (even far)beyond the virial radius. TJ03b,c carefully verified that employing
the expressions (12) and (13) in the real-space halo model is essential to achieve model accuracy as well as consistency with theFourier-space halo model well investigated in the literature (e.g.
Seljak 2000).
Throughout this paper we employ the halo boundary r180 , notthe virial radius, as stated in Section 3.1. For this setting, we haveto replace parameter c used in F(x) and G(x) in TJ03b,c with theratio of the scale radius to r180 , c180 . The relation between c180
and c is given by c180 = cr180 /rvir . Note that, even if we use
the virial boundary, the results shown in this paper are almost
unchanged.
Given the convergence and shear profiles for a halo of mass M, themagnification profile is given by µM (θ) = |(1−κM (θ))2 −γM2 (θ)|−1 .
This equation implies that µM becomes formally infinity at somecritical radius when the denominator becomes zero. The radius in
the lens plane is called the critical curve, while the corresponding
curve in the source plane is the caustic curve.
Any lensing NFW halo inevitably provides finite critical radius, if
we have an ideal angular resolution. This is because the convergenceκM varies from zero to infinity with changing radius from θ180 to 0,
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Magnification statistics

953

dominant contribution on small non-linear scales (see equations 19
and 20 in TJ03c): χHξµ1h (θ) =
0

d2 V
dχ
dχd



 ∞

dMn(M)

dφ

 2π

×

0

dϕφ[µ M (φ) − 1][µ M (|φ + θ|) − 1].

(16)

0

Figure 1. The tangential critical radius of the NFW profile against halo
masses. The solid and dashed curves are the results for lensing haloes at
redshifts of zl = 0.4 and 0.8 for source redshifts of zs = 1 and 3, respectively.
Even if we consider a lower lens redshift, the result does not change greatly,
as shown by the dot-dashed curve.

while the shear remains finite over the range (see fig. 2 in TJ03c).
Fig. 1 plots the tangential critical radius for a lensing NFW halo
against the mass. Note that an NFW profile produces two criticalradii – an outer curve causes a tangentially deformed image aroundit, while the inner one causes a radially deformed image. The solidand dashed curves are the results for haloes at redshifts zl = 0.4 and
0.8 for source redshifts of zs = 1.0 and 3.0, respectively. We can seethat the critical radius has a strong dependence on halo masses and islarger for zs = 3 than for zs = 1 due to the greater lensing efficiency.
These critical curves do not largely change even if we consider alower lens redshift than the peak redshift, as shown by the dot-dashedcurve. Even massive haloes with M ∼ 1015 M provide the criticalradii of  0.1 arcmin. The scale is below relevant angular scales forthe magnification statistics of our interest. However, this does notmean that the non-linear magnification correction to the correlation
function appears only on scales 1 arcmin. Rather, modest nonlinear magnifications (µ  2) lead to the strong impact, becausesuch magnifications have a larger cross-section, as will be shown indetail.

Here we have introduced the polar coordinate d2 φ = φ dφ dϕ,d2 V/dχ d = dA2 (χ) for a flat universe and we can set the separation vector θ to be along the first axis from statistical symmetry,
thereby |φ + θ| = (φ 2 + θ 2 + 2φθcos ϕ)1/2 . We have assumed auniform distribution of sources on the sky and ignored a probability of multiple images and an increase or decrease in sampling of
the images due to the lensing itself (which is a higher-order correction and can be safely neglected as shown in Hamana 2001).
The equation above implies that the one-halo term contribution is
given by summing lensing contributions due to haloes along the lineof sight weighted with the halo number density on the light cone.
Note that the integration range of d2 φ is taken as the infinite area,taking into account the non-local property of the shear field that is
non-vanishing at radius 
outside the halo boundary. In practice, setting the upper bound of dφ to be three times the projected radiusθ180 gives the same result, to within a few per cent. The validity of
the real-space halo model formulation was carefully investigated in
TJ03b,c.
As discussed in Section 3.2 and shown in Fig. 1, an NFW profile always provides finite critical curves, where the magnification
formally becomes infinity. Therefore, to make the halo model prediction, we introduce a magnification
cut-off µmax in the calculation
– the integration range of d2 φ is confined to the region satisfying
the condition µM  µmax for a given halo. This procedure is somehow similar to what is done in the measurement from ray-tracing
simulations, where a masking of high magnification events is employed to avoid a significant statistical scatter (see, for example,
Ménard et al. 2003; Barber & Taylor 2003). Thus, the halo modelallows for a fair comparison of the prediction with the simulation
result. However, note that the procedure taken ignores the lensing
projection effect for simplicity; exactly speaking, the magnification
should be given by the lensing fields between source and observer,
not by those of the individual halo.
Similarly, based on the real-space halo model, the two-halo term
can be expressed as ∞ξµ2h (θ) =dχ
0

3.3 Real-space halo model approach

×



d2 V
dχd





dM n(M)d φ[µ M (φ) − 1]2

In the following, we construct the halo model method to compute themagnification-induced correlation function. First, we simply consider the 2PCF defined as× b(M)b(M  )ξµ (θ ) ≡ [µ(φ) − 1][µ(φ + θ) − 1] ,× J0 (l|θ − φ − φ |),(14)

where µ − 1 is the magnification fluctuation field. This 2PCF is
observable from size fluctuations of distant galaxy images, as discussed in Section 2.2.
From a picture of the halo model, ξµ (θ ) can be expressed as a sumof correlations between the magnification fields within a single halo(one-halo term) and between two different haloes (two-halo term):ξµ (θ ) = ξµ1h (θ ) + ξµ2h (θ ).

(15)

It is straightforward to extend the real-space halo model developed
in TJ03a,b,c to compute the one-halo term contribution, which has
C2003 RAS, MNRAS 346, 949–962

dM  n(M  )

d2 φ [µ M  (φ  ) − 1]



l
ldl L
P k=
;χ
2π
dA



(17)
L

where J 0 (x) is the zeroth-order Bessel function and P (k) is thelinear mass power spectrum at epoch χ as given by PL (k; χ ) =D2 (χ)P(k; t0 ). The term including PL (k) in the third line on the righthand side (rhs) denotes the angular 2PCF between different haloesof masses M and M, which is derived using Limber’s equationand the flat-sky approximation (e.g. Blandford et al. 1991; MiraldaEscude 1991; Kaiser 1992). We similarly introduce the maximum
magnification cut-off in the two-halo term calculation as in the onehalo term. The equation above means that we have to perform aneight-dimensional integration to obtain the two-halo term, which

954

M. Takada and T. Hamana

is computationally intractable. For this reason, we employ an approximated way to be valid when angular separation between thetwo haloes taken is much greater than their angular virial radii. This
leads to a simplified expression of the two-halo term: ∞ξµ2h (θ ) ≈

dχ
0



×

d2 V
dχ d


×

2


φ dφ2π(µ M (φ) − 1)dMb(M)n(M)



l
ldl L
;χ
P k=
2π
dA (χ )

J0 (lθ).

(18)

This allows us to obtain the two-halo term by a 3D integration,
because the integrations of the second and third lines on the rhs canbe performed separately before carrying out the χ -integration. It is
worth checking the consistency of the two-halo term above with thelimiting case ξµ ≈ 4ξκ , which is valid in the weak lensing limitµ ≈ 1 + 2κ for κ, γ  1. In this limit, the term contained in thesecond line on the rhs of equation (18) can be rewritten as






≈



dMb(M)n(M)d2 φ2κ M (φ)

= 2W (χ)dA−2 (χ )dMb(M)n(M)

M
,
ρ̄0

(19)

where the second equality is derived from equations (25) and (28)
in TJ03b. Therefore, substituting this result into the two-halo term(18) yields

 ∞ξµ2h (θ ) ≈ 4

dχ W 2 (χ )dA−2 (χ )

0


×

dMb(M)n(M)


×



M
ρ̄0

2

l
ldl L
P k=
;χ
2π
dA (χ )

1
.
(21)
θ
This argument holds independently of details of the lensing mass
distribution, although the proportional coefficient does depend on
the mass distribution.
To keep the generality of our discussion, let us consider a correlation function between the magnification field and some cosmic
fields expressed asµ∝ξ (θ; p) = [µ(θ 1 )] p f (θ 2 )|θ1 −θ2 |=θ ,d2 φ(µ M (φ) − 1)dMb(M)n(M)In the following, we clarify how the magnification statistics depends
upon large magnification events (µ  1).
Meanwhile, we restrict our discussion to point-like sources for
simplicity. High magnifications (µ  1) arise from images in thevicinity of the critical curve that is caused by an intervening massconcentration, such as haloes. For any finite lens mass distribution,
the critical curve must form a closed non-self-intersecting loop.
Based on the catastrophe theory, it was shown in Blandford &Narayan (1986; also see Chapter 6 in Schneider et al. 1992) thatthe magnification of an image at a perpendicular distance θ fromthe (fold) critical curve scales asymptotically asJ0 (lθ)= 4ξκ2h (θ ).

(20)

The two-halo term (18) is thus reduced to four times the two-haloterm of the convergence 2PCF in the weak lensing limit, as expected.
However, the important point of the two-halo term (18) is that it cancorrectly account for the contribution of non-linear magnifications(µ  2) on large scales.
Replacing P2h (k) in equation (20) with a model of the non-linear
mass power spectrum leads to the conventional method for predicting 4ξκ (θ ) as an estimator of ξ µ (θ ), as employed in the literature(e.g. Dolag & Bartelmann 1997; Sanz et al. 1997; Jain et al. 2003).
4 A S Y M P T OT I C B E H AV I O U R O F
M AG N I F I C AT I O N S TAT I S T I C S
F O R H I G H M AG N I F I C AT I O N S
As stated in the preceding section, we need to introduce a maximum
magnification cut-off in the model prediction to avoid the contribution from a formally emerged infinite magnification. In practice,strong lensing events of µ  1, which are identified by multiple images or largely deformed images, can be removed from the sampleof the magnification statistics. However, without clear signatures, it
is hard to make the distinct selection and therefore modest magnification events (δµ  1) are likely included in the sample for theblind analysis, because the magnification is not a direct observable.

(22)

where p is an arbitrary number. The field f is allowed to be constituted from any cosmic fields which are correlated with the magnification field. Therefore, the following argument holds for high-ordermoments beyond the two-point correlations if we take products ofthe cosmic fields for f , e.g. f = δ(θ 2 )δ(θ 3 ).
Suppose that an intervening halo provides the critical curve in thelens plane for a given source redshift, as this is the case for an NFW
halo (see Fig. 1). Then, let us consider how high magnificationsin the vicinity of this critical curve contribute to the magnification
statistics. By introducing an upper bound on the magnification fields
as µ  µmax , we can address how the magnification statistics dependon the cut-off µmax and what is the asymptotic behaviour for thelimiting case µmax → ∞. From equation (21), the cut-off µmax
corresponds to a lower limit on the perpendicular distance fromthe critical curve, say θ   ( → 0 corresponds to µmax →∞). As can be seen from equations (16) and (18), a picture of thehalo model leads us to compute the high magnification contribution
to the magnification-induced correlation like equation (22) by the
integrationξ (θ; p, µmax ) ∼d2 s∂Sc ,µµmax

1
f (|s + θ|),|s| p(23)

where s is the perpendicular distance from the critical curve andthe integration range is confined to the area subject to the conditionµ  µmax . The integration (23) results in one-dimensional integration for high magnifications around the critical curve, in analogy
with equation (5.16) in Blandford & Narayan (1986) to derive theasymptotic, integral cross-section for the strong lensing events that
produce multiple images.7 Hence, the leading order contribution ofµmax can be expressed asξ (θ; p, µmax ) ∼


1− p

1/µmax ,ln(µmax ),µ p−1 ,
max

p<1
p=1

(24)

p > 1.

7 The discussion in this paper, as well as in Blandford & Narayan (1986),

employs the assumption that asymptotic dependence of the magnification
statistics on high magnifications is mainly due to images around the foldcaustics and therefore ignores the contribution from the cusp caustic. Thisis likely to be a good approximation as shown in Mao (1992).
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955

where we have assumed that variation in the field f does not largely
change for the relevant integration range. The equation above leads
to an intriguing consequence; the amplitude of the magnification
correlation is finite for p < 1 for the limiting case µmax → ∞,while it diverges for p  1. Thus, the statistics with p < 1 is practically advantageous in that it is insensitive to the uncertainty ofwhich magnification cut-off should be imposed for a given sample.
Furthermore, the asymptotic behaviour does not explicitly dependon the separation angle θ and therefore it holds even for large θ.
This means that the divergence ξ → ∞ for p  1 formally occurs
even on degree scales, which is opposed to a naive expectation thatthe weak lensing approximation is safely valid on these scales. It isalso worth stressing that this behaviour is expected to hold for any
lensing mass distribution once the critical curve appears, although
the proportionality coefficient of ξ should depend on details of themass distribution. These will be quantitatively tested by the halomodel prediction as well as by the ray-tracing simulation.
In reality, a finite source size imposes a maximum cut-off on
the observed magnification and thus the infinite magnification does
not occur, even if a source sits on the caustic curve in the source
plane (see Chapters 6 and 7 in Schneider et al. 1992; Peacock 1982;
Blandford & Narayan 1986). For example, if the source is a circular
disc with radius rS and uniform surface brightness, the maximum√
magnification is given by µmax ∝ 1/ rS . Therefore, to develop an
accurate model prediction requires a knowledge of unlensed source
properties such as the size and the surface brightness distributionin addition to modelling the lensing mass distribution. In particular, this could be crucial if we consider the magnification statistics (22) with p  1, because it is sensitive to large magnification
events.

5 R E S U LT S
5.1 Ray-tracing simulations
To test the analytical method developed in Section 3, we employ the
ray-tracing simulations. We will use the simulation for the current
concordance CDM model with m0 = 0.3, λ0 = 0.7, b0 =0.04, h = 0.7 and σ8 = 0.9 (Ménard et al. 2003; Hamana et al., in
preparation; TJ03c). The N-body simulations were carried out by
the Virgo Consortium8 (see alsoYoshida, Sheth & Diaferio 2001),
and were run using the particle–particle/particle–mesh (P3 M) codewith a force softening length of lsoft ∼ 30 h−1 kpc. The initial matter
power spectrum was computed using CMBFAST (Seljak & Zaldarriaga
1996). For the analytical model, we approximate the initial condition
to use the CDM transfer function given by Bardeen et al. (1986)
with the shape parameter in Sugiyama (1995) for simplicity. The
N-body simulation employs 5123 CDM particles in a cubic box of479 h−1 Mpc on a side, and the particle mass of the simulation ismpart = 6.8 × 1010 h−1 M .
The multiple-lens plane algorithm to simulate the lensing maps
from the N-body simulations is detailed in Jain et al. (2000) andHamana & Mellier (2001). We will use the output data for source
redshifts of zs = 1 and 3 for the following analysis. The simulated
map is given on 10242 grids of a size of θgrid = 0.2 arcmin; the area
is s = 11.7deg2 . The angular resolution that is unlikely affectedby the discreteness of the N-body simulation is around 1 arcmin(Ménard et al. 2003; TJ03c).

8 See http://www.mpa-garching.mpg.de/Virgo/for details.
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2003 RAS, MNRAS 346, 949–962

Figure 2. The PDF of the magnification measured from the ray-tracing
simulations for zs = 1 and 3. To clarify the angular resolution, we show thetwo results of using different smoothing scales, which were used to avoid theartificial discrete effect of the N-body simulations. The comparison manifests
that high magnification events are sensitive to smaller structures that are
relevant for the smoothing. The dashed line shows an asymptotic behaviourof the PDF for high magnification events, PDF ∝ µ−2 , as theoretically
expected (see the text for more detail).

5.2 Probability distribution function of the magnification: theangular resolution of the simulations
Universal properties of the critical curve, as demonstrated in Section 4, lead to an asymptotic behaviour of the probability distributionfunction (PDF) of large magnification events, irrespective of detailsof the lensing mass distribution: PDF(µ) ∝ µ−2 (Peacock 1982; Vietri & Ostriker 1983; Blandford & Narayan 1986; Schneider 1987;
Hamana, Martel & Futamase 2000; also see Chapters 6, 11 and 12
of Schneider et al. 1992). Note that the PDF is defined in the image
plane, while the PDF becomes µ−3 if we define it from the sample
of sources in the source plane. We use this property to investigate
the angular resolution of the ray-tracing simulation.
Fig. 2 shows the magnification PDF measured in the simulations for source redshifts of zs = 1 and 3, respectively. To compute the PDF, we accumulate the counts in a given bin of µ from36
 realizations and then normalize the PDF amplitude to satisfydµPDF(µ) = 1 over the range of µ measured. The PDF has askewed distribution; most events lie in demagnification of µ < 1956

M. Takada and T. Hamana

and rare events have high magnifications with µ  1 having a longtail. These reflect an asymmetric mass distribution in the large-scale
structure as expected from the CDM scenario – the underdense region can be seen preferentially in the void region with a typicalsize ∼10 Mpc, while the highly non-linear structures appear in dark
matter haloes on scales Mpc. As can be seen, the simulation ofzs = 3 displays more pronounced evidence of asymptotic dependence PDF(µ) ∝ µ−2 for high magnifications (µ  10) than the
result for zs = 1.
To more explicitly clarify the resolution issue of the simulations,
we show the two results of the different smoothing scales that wereused in making the projected density field to suppress the discreteness effect of the N-body simulations [see Ménard et al. 2003 andHamana et al. (in preparation) for more details]. They are called‘small-scale smoothing’ (solid curve) and ‘large-scale smoothing’(dotted curve), respectively. The former is expected to have an angular resolution around 1 arcmin as discussed in Ménard et al. (2003)
and in TJ03c, while the latter employs a smoothing scale two times
larger than the former. The effect of the large-scale smoothing isthat it more smoothes out smaller-scale structures of the mass distribution that are resolved by the small-scale smoothing simulation.
The comparison manifests that occurrence of high magnification
events (µ > 1) is very sensitive to the small-scale structures. For
this reason, we will employ the small-smoothing simulation in thefollowing, because our interest is to clarify the non-linear magnification effect on the magnification statistics. This result also implies
that simulations with higher resolution could further alter the PDF
shape especially at high magnification tail.
5.3 The 2PCF of lensing magnification
We now turn to an investigation of the magnification 2PCF, ξµ ≡δµδµ, as it is possible to observe from size fluctuations on distantgalaxy images (see Section 2.2). Fig. 3 shows the comparison of
the halo model prediction (solid curve) with the measurement fromsimulations (triangle symbol) for source redshift zs = 1. Note thatthe error bar in each bin denotes the sample variance for a simulatedarea of 11.7 deg2 , which is computed from 36 realizations, and theerrors in neighbouring bins are highly correlated. In this and following results, we mainly employ the maximum magnification cut-offµmax = 8 in the halo model prediction as well as in the simulation
result. If we ignore the shear contribution to the magnification (1),this cut-off corresponds to µ ≈ 1 + 2κ = 2.3 for the weak lensing
approximation. The cut-off value is chosen so that strong lensing
events are removed from the analysis, because such events likely
have greater magnification µ  10 (M. Oguri, private communication). Fig. 2 shows that this cut-off leads us to exclude the events ina high magnification tail of the PDF.
Fig. 3 shows that the halo model prediction matches the simulation result well. The one-halo term provides a dominant contribution to the total power on small scales  3 arcmin, while thetwo-halo term eventually captures the larger-scale signal (see fig. A1of TJ03c). It is worth noting that the shear field in µ (see equation 1)
contributes to the 2PCF amplitude by ∼10 per cent over the scalesconsidered. To make clear the importance of the non-linear magnification contribution (δµ  1), the dashed curve and the squaresymbol are the halo model prediction and the simulation result for
the weak lensing approximation ξµ ≈ 4κκ = 4ξκ . For this case,4ξκ can also be computed from the fitting formula of the non-linear
mass power spectrum recently proposed by Smith et al. (2003),
which demonstrates another test of the accuracy of the halo model
as well as of the simulation.

Figure 3. The 2PCF of the magnification field against the separation angle
for the CDM model and source redshift zs = 1. The solid curve shows
the halo model prediction, while the triangle symbol is the simulation result.
In most of this paper, the maximum magnification cut-off µmax = 8 isemployed (see text for details). The error bar denotes the sample variance fora simulated area of 11.7 deg2 . For comparison, the weak lensing predictions,leading to ξµ ≈ 4κκ, are shown by the halo model (dashed curve), thesimulation (square symbol) and the fitting formula (dot-dashed curve) of
Smith et al. (2003). Note that the simulation result is slightly shifted in thehorizontal direction for illustrative purposes. The lower panel explicitly plots
the contribution of non-linear magnifications, ξµ /4ξκ −1, for the halo model
prediction and the simulation result.

The lower panel explicitly plots the relative difference, ξµ /4ξκ −1. The simulation result is computed from the mean values ofξµ and 4ξκ , and we do not plot the large error bar for illustrative
purpose. The correction of the non-linear magnification amounts to30 per cent at θ  2 arcmin, and the non-negligible contributionof ∼10 per cent still remains even on large scales  10 arcmin. Ourmodel of the two-halo term (18) correctly captures the non-linear
effect seen in the simulation on the large scales. This large-scalecorrection is surprising, because it is naively expected that the weaklensing approximation is valid at these scales. Ménard et al. (2003)
also showed that the non-linear correction on the large scales can be
fairly explained taking into account the higher-order terms O(κ 2 ) in
the Taylor expansion of µ, although the method ceases to be accurate on small scales 5 arcmin. One advantage of the halo model isthat it allows us to explicitly introduce the maximum magnification
cut-off in the calculation, which allows a fair comparison with thesimulation and probably with the actual observation. In other words,
the results shown depend upon the cut-off value employed. If we use
the cut-off values of µmax = 2 and 100, the deviation, ξµ /4ξκ − 1,becomes ∼20 and ∼40 per cent at θ = 1 arcmin, respectively (alsosee Fig. 5).
Fig. 4 shows the result for zs = 3, as in the previous figure. It isclear that the non-linear magnification contribution leads to significant enhancement in the amplitude of the magnification correlation
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2003 RAS, MNRAS 346, 949–962

Magnification statistics

Figure 4. As in the previous figure, but for zs = 3. For higher source
redshifts, the non-linear magnification effect becomes more significant as
expected.

relative to the weak lensing prediction. The enhancement is 40
per cent at θ  2 arcmin. The comparison with the previous figure manifests that sources at higher redshifts are more affected by
the non-linear magnification, as the sources have more chance toencounter intervening haloes. It is worth noting that, if we do not
apply any masking of high magnification events in the simulation,
the statistical error in each bin becomes very large, which indicatesthe presence of events with µ  1 in some realizations as shown in
Fig. 2.
5.4 A quantitative test of the asymptotic behaviourof the magnification statistics
In what follows we quantitatively test the asymptotic dependenceof the magnification statistics on large magnifications (µ  1),
as derived in Section 4. For this purpose, we consider three casesof p = 0.5, 1 and 1.5 for the magnification 2PCF parametrized asξ = δµp δµp , with source redshift zs = 3. Fig. 5 shows how the2PCF amplitude depends on the magnification cut-off µmax used in
the halo model predictions and the simulation result. The separation
angle of θ = 1 arcmin is considered and the curves are normalized
by the predictions from the weak lensing approximation. The scaleθ = 1 arcmin is chosen based on the fact that the scale is in the nonlinear regime and unlikely affected by the angular resolution of thesimulation (TJ03c). First, we can see that even the most conservative
choice of µmax = 2 leads to a decent difference between the correct
treatment and the weak lensing approximation. The consequence
derived in Section 4 is that the 2PCF amplitudes for p = 0.5, 1.0 and
1.5 have the dependences on µmax given as ξ ∝ µ−0.5
max , ln(µmax ) andµ0.5
max for µmax  1, respectively. It is obvious that this consequenceis verified by the halo model prediction as well as by the simulation
result for µmax  10. Although the halo model results for α = 1.0
and 1.5 display a bend at µmax ≈ 30, we have found that this isdue to high magnifications between the radial and tangential criticalC

2003 RAS, MNRAS 346, 949–962

957

Figure 5. Dependences of the magnification 2PCF amplitude on the maximum magnification cut-off used in the evaluations. For the magnification
correlation parametrized as δµp δµp , the three panels show the results for
p = 1, 0.5 and 1.5, respectively. The source redshift zs = 3 and the separation angle θ = 1 arcmin are considered. All the curves are normalized by thepredictions derived from the weak lensing approximation (µ ≈ 1 + 2κ). As
discussed in Section 4, it is expected that the 2PCF amplitude diverges forp  1 for the limiting case µmax → ∞, while it remains finite for p < 1 (seeequation 24). This is verified by the halo model as well as by the simulation.

curves in NFW haloes. Most importantly, the 2PCF amplitude for
p = 0.5 has a well-converged value for µmax  5; the amplitudechanges by less than 10 per cent over µmax = [10, 1000]. Therefore,
the statistics with p < 1 has a practically great advantage because
it is little affected by the uncertainty in specifying the maximum
magnification cut-off in the analysis.
Finally, we note that the results shown above are unchanged evenif we consider the cross-correlation δµp δ 2D , where δ 2D is the projected density fluctuation field (e.g. see equation 24), because theasymptotic behaviour is determined by the power of µ entering into
the general correlation function µp f , as derived in Section 4.
5.5 Application to QSO–galaxy cross-correlation
In this section, we consider an application of the halo model to
the QSO–galaxy cross-correlation. The angular fluctuation field ofgalaxies on the sky is a projection of the 3D galaxy fluctuation
field δ g along the line of sight, weighted with the redshift selection
function f g (z) of the galaxy sample

 ∞δn g (θ̄) =dz f g (z)δ g (dA (z)θ, z),
0

(25)

∞

where f g (z) is normalized as 0 dz f g (z) = 1. Throughout this paper,
we employ β 

βz 2
z
exp −
f g (z) dz = 3z0z 0 (3/β)

dz,

(26)

958

M. Takada and T. Hamana

Figure 6. The cross-correlation function between the magnification field and the projected density fluctuation field for zs = 1 (left panel) and 3 (right panel),as in Fig. 3. To obtain the projected density field, we assume the redshift selection function given by equation (26).

with β = 1.5 and z0 = 0.3. This model leads to the mean redshift asz mean = dz z f g (z) = 0.45 and roughly reproduces the actual distribution of galaxies in the redshift galaxy catalogue (e.g. Dodelson
et al. 2002).
However, the galaxy fluctuation field δ g is not straightforward tomodel, because the galaxy formation is affected by complex astrophysical processes in addition to the gravitational effect. Recently,
Jain et al. (2003) developed a sophisticated description of the magnification correlations based on the halo model as well as the semianalytical galaxy formation model. In particular, it was shown that
it is crucial to account for a realistic model to describe how galaxies
populate their parent halo, the so-called halo occupation number,to make the accurate model predictions on arcmin angular scales.
The halo occupation number strongly depends on types of galaxies such red or blue galaxies. We here address how the non-linear
magnification further modifies the model prediction.
Before going to this study, we consider the cross-correlation between the magnification field and the dark matter distribution, whichcorresponds to an unrealistic case that the galaxy distribution exactly
traces the underlying mass distribution; δg = δ. This investigation is
aimed at clarifying how the non-linear magnification effect remains
after inclusion of a realistic model of the galaxy clustering, from thecomparison of the results with and without the galaxy bias model. In
addition, in this case we can compare the model prediction with thesimulation result that is computed from the same N-body simulation
we have used. Extending the method presented in Section 3 leads
to the one-halo term contribution to the cross-correlation

 χH

1h
ξµδ
(θ ) =

dχ
0

 2π

d2 V
dz
f g (z)
dχd
dχ





dMn(M)



M
ρ̄0

 ∞

0

 χH
1h
ξµg
(θ) =

dχ
0

 ∞

×

d2 V
dz
f g (z)
dχd
dχ

 2π

dMn(M)

dϕφ µ2.5s−1(φ) − 1M

dφ0Ng (M)
n̄ gal

0× M (|φ + θ|),

dφ

(28)

0

dϕφ µ2.5s−1(|φ + θ|) − 1  M (φ),
M

×where  M (x) is the normalized projected density of the NFW profile
given by equation (26) in TJ03b. Similarly, one can derive the twohalo term of ξµδ , as done by equation (18).
In Fig. 6 we show the results for source redshifts of zs = 1 (leftpanel) and 3 (right panel), as in Figs 3 and 4. Note that both theresults employ the same redshift selection function (26) to obtain
the projected density field. We simply assumed a special case ofs = 4/5 for the magnitude slope for the unlensed QSO number
count. From the comparison with Figs 3 and 4, it is clear that the
non-linear magnification contribution is weakened, due to the single
power of µ entering into the two-point correlation compared to themagnification 2PCF. Nevertheless, it should be stressed that the nonlinear correction has significant contributions of 15 and 25 percent at θ  1 arcmin for zs = 1 and 3, respectively.
Next, we consider a model of the QSO–galaxy correlation thattakes into account both the galaxy bias and the non-linear magnification effect. To do this, we use the halo occupation number N g (M)to describe how many galaxies populate their parent halo of a given
mass M, in an average sense (e.g. Seljak 2000; Guzik & Seljak 2002;
Jain et al. 2003; TJ03b; Cooray & Sheth 2002). Simply replacingM/ρ̄0 in equation (27) with Ng (M)/n̄ gal leads to the one-halo termof the QSO–galaxy cross-correlation(27)

where n̄ gal is the average number density at epoch z defined asn̄ gal = dMn(M)Ng (M). The cross-correlation thus depends on
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959

Figure 7. The halo model predictions for the angular cross-correlation between the magnification and the foreground galaxy distribution, where the galaxy
clustering is modelled by the halo occupation number (31). For comparison, the dot-dashed curve is the result for the cross-correlation between the magnificationand the projected dark matter distribution as shown in the previous figure. The non-linear magnification correction remains to enhance the cross-correlation
amplitude relative to the weak lensing approximation, even if the realistic model of the galaxy clustering is included.

the first moment of Ng (M).9 Note that, on the other hand, the twopoint correlation of galaxies depends on the second moment, andit contains somehow uncertainty in modelling the subPoissonian
process in the regime of Ng (M) < 1.
As stressed in Guzik & Seljak (2002) and Jain et al. (2003), it
is probably accurate to assume that one of the Ng  galaxies in ahalo sits at the halo centre and this has a decent impact on themodel predictions. On the other hand, we assume that the other(Ng  − 1) galaxies follow the dark matter distribution within thehalo. Following the method of Jain et al. (2003), the part of theintegrand function in the one-halo term (28) can be expressed as ∞Ng (M)

 2π



0

0

= χ −2 µ2.5s−1(θ ) − 1
M

 ∞

+

 2π









dϕ φ µ2.5s−1(φ) − 1  M (|φ + θ|)
M

dφ
0



dϕφ µ2.5s−1(φ) − 1  M (|φ + θ|)
M

dφ

0

× [Ng (M) − 1],(29)for Ng (M)  1 and(θ ) − 1 Ng (M),= µ2.5s−1
M(30)

for Ng (M) < 1. Substituting these equations into equation (28)leads to the halo model prediction for the QSO–galaxy crosscorrelation.
9 This is also the case for galaxy–galaxy lensing as shown in Guzik & Seljak(2002).
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2003 RAS, MNRAS 346, 949–962

To complete the model prediction, we need an adequately accurate
model of the halo occupation number Ng (M). We employed themodel in Jain et al. (2003), which was derived from the GIF N-bodysimulations, coupled to a semi-analytical galaxy formation model(Kauffmann et al. 1999). The simulation result of Ng (M) is wellfitted by the functional formN g (M) =

M
M0

α





+ A exp −A0 (log10 (M) − M B )2 .

(31)

The parameter values are labelled as ‘z = 0.06’ taken fromtable 1 in Jain et al. (2003), which reproduces the measurements
for total (blue plus red) galaxies at z = 0.06 in the GIF simulations. In the following, we employ a lower mass cut-off ofM  1011 h−1 M and ignore the redshift evolution of Ng (M)for simplicity. This model leads to the galaxy bias parameterb̄gal = (1/n̄ gal ) dMn(M)b(M)Ng (M) = 1.2 at z = 0 in thelarge-scale limit, thus reflecting the fact that the modelled galaxies
are biased objects relative to the dark matter distribution.
Fig. 7 shows the model predictions for the QSO–galaxy crosscorrelation, as in the previous figure. For comparison, the dot-dashedcurve is the result for the cross-correlation between the magnification and the projected dark matter distribution in the previous
figure. The comparison of the solid and dot-dashed curves manifests that the realistic model of galaxy bias largely modifies the
cross-correlation, and the galaxy bias cannot be described by asimple linear bias on the small angular scales (Jain et al. 2003).
It is also clear that the non-linear magnification correction is 10–
25 per cent on arcmin scales. Therefore, this result implies thatan inclusion of the non-linear effect will be necessary to make an

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Figure 8. The dependences of the magnification 2PCF, δµ0.5 δµ0.5 , on the halo profile parameters. The left panel shows the halo model predictions for thehalo concentration parameter c0 = 3, 9 and 15 (from bottom to top), while the right panel shows the results with α = 0, 1 and 2 for the inner slope parameter ofthe generalized NFW profile (see text for more details). For comparison, the dashed curves are the corresponding results for the convergence 2PCF. Note thatthe magnification 2PCF considered becomes equivalent to the convergence 2PCF in the weak lensing limit as δµ0.5 ≈ κ. The lower panel shows the difference
relative to the result with our fiducial model of α = 1 and c0 = 9.

unbiased interpretation of the precise measurement expected fromforthcoming massive surveys such as the CFHT Legacy Survey andthe SDSS.
5.6 Sensitivity of the magnification statistics
to the halo profile properties
As discussed above, some of the useful cosmological information
extracted from the QSO–galaxy correlation measurement is information on the halo occupation number of galaxies, which in turnprovides a clue to understanding galaxy formation in connection to
the dark matter halo properties (see Jain et al. 2003, for details). We
here demonstrate another possibility of the magnification statistics(especially measured via galaxy size fluctuations) to address the following questions. What can we learn from the measurements? How
is this method complementary to the established cosmic shear thatprobes the correlations of the convergence or shear fields (κ or γ )?
To examine this, we focus on the non-linear relation between themagnification and the cosmic shear fields, as given by equation(1). The non-linear effect is more pronounced on smaller scales,
as has so far been shown. Future massive surveys promise to measure the magnification statistics as well as the cosmic shear evenon subarcmin scales (Jain 2002; TJ03c; Jain et al. 2003). Withina picture of the halo model, the subarcmin correlation function
is quite sensitive to the halo profile properties (TJ03b,c) and themeasurement can be potentially used to constrain the properties,
if the systematics is well under control. Hence, we here investigate the dependence of the magnification 2PCF on the halo profile parameters, the halo concentration and the inner slope of thegeneralized NFW profile. These parameters are still uncertain observationally and theoretically and have information on the dark
matter nature as well as properties of highly non-linear gravitational clustering on 1 Mpc. Following TJ03c, we consider theparametrization given as c(M, z) = rvir /rs = c0 (M/M∗ )−0.13 andρ(r) ∝ r−α (1 + r/rs )−3+α , respectively. Our fiducial model so far
used is given by (c0 , α) = (9.0, 1.0). For cases α = 0, 1 and 2
we can derive analytical expressions for the convergence and shear
profiles from which we can also compute the magnification profile(the expressions of the convergence fields are given in appendix B
in TJ03c). Note that in what follows we employ the virial boundary condition. The relevant angular scales are below the angularresolution of N-body simulations we have used.
The left panel of Fig. 8 shows the halo model prediction forthe magnification 2PCF with varying the halo concentration, while
the right panel shows the results with varying α. Here we consider
δµ0.5 δµ0.5 , because it is less sensitive to high magnification events(µ  1) and therefore observationally more robust (see Fig. 5). In
the weak lensing limit, the correlation can be approximated by theconvergence 2PCF (dashed curves), which is measured by the cosmic shear measurement because δµ0.5 ≈ κ. Therefore, the difference between the solid and dashed curves reflects the contribution
from the non-linear magnifications δµ  1. Haloes with masses
M  1013 M provide a dominant contribution of 80 per cent to
the total power over a range of non-linear scales 0.1–3 arcmin (see,for example, fig. 14 in TJ03c). We can see that the magnification
2PCF has stronger sensitivity to the halo concentration and depends
on the inner slope in a different way from the convergence 2PCF.
TJ03c pointed out that the cosmic shear measurement introduces aC2003 RAS, MNRAS 346, 949–962

Magnification statistics
degeneracy in determining these halo profile parameters, even providing the accurate measurement (see figs 16 and 17 in TJ03c). The
results in Fig. 8 thus indicate that a joint measurement of the magnification statistics and the cosmic shear can be used to improve theparameter determinations. Finally, one caution we make is that themagnification 2PCF for the profile with α = 2 is more amplified
by an increase of the maximum magnification cut-off µmax than theother values of α and thus is sensitive to the selection effect.
6 DISCUSSION
In this paper, we have used the real-space halo approach to computethe magnification correlation function without employing the weaklensing approximation µ ≈ 1 + 2κ + O(κ 2 ). It has been shown thatthe correction due to the non-linear magnification (δµ  1) leads
to significant enhancement in the correlation amplitude relative tothe weak lensing approximation (see Figs 3–7). The correction ismore important as we consider the correlation function for sources
at higher redshifts and on smaller angular scales, where the weaklensing approximation ceases to be accurate. Thus, it is necessary to
account for the non-linear contribution in the theoretical model inorder to extract unbiased, cosmological information from the precise
measurement expected from forthcoming and future surveys. The
encouraging result shown is that the halo model prediction remarkably well reproduces the simulation result over the angular scales
we consider.
We also developed the model to predict the QSO–galaxy crosscorrelation by incorporating the realistic model of the halo occupation number of galaxies into the halo model (see Section 5.5).
The primary cosmological information provided from the measurement is constraints on the halo occupation number, as shown in
Jain et al. (2003); also see Guzik & Seljak (2002). In particular, the
QSO–galaxy correlation can be used to directly constrain the first
moment of the halo occupation number, compared to the two-point
correlation of galaxies that probes the second moment. Exploring
the halo occupation is compelling in that it provides useful information on the galaxy formation and the merging history in connection
with the dark matter halo properties. We have shown that the nonlinear magnification amplifies the cross-correlation amplitude by
10–25 per cent on arcmin scales. The method of this paper therefore
provides the accurate model prediction that accounts for both the
non-linear magnification correction and the realistic galaxy bias.
We found that the magnification statistics can be used to extract
cosmological information complementary to that provided from thecosmic shear measurement. We have demonstrated that the joint
measurement on angular scales 3 arcmin could be used to precisely constrain the halo profile properties (see Fig. 8). This possibility would open a new direction in using the magnification statistics
as a cosmological probe beyond determination of fundamental cosmological parameters (Bartelmann 1995; Bartelmann & Schneider
2001; Ménard & Bartelmann 2002; Ménard et al. 2003). Exploring
the halo profile properties with gravitational lensing will be a direct
test of the CDM scenario in the highly non-linear regime Mpc,
because alternative scenarios have been proposed in order to reconcile the possible conflicts between the CDM predictions and theobservations on small scales (Spergel & Steinhardt 2000).
In most results shown, we employed the maximum magnification
cut-off µmax = 8 for the halo model predictions as well as for thesimulation results, because the choice likely removes strong lensing
events (µ  10) from the analysis. Even if we employ the smaller
value, the qualitative conclusions derived are not largely changed,
as can be seen from Fig. 5. Observationally, a strong lensing event isC

2003 RAS, MNRAS 346, 949–962

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easily removed from the sample of the magnification statistics, if it
accompanies multiple images or largely deformed images. However,
without the clear signature, it is relatively difficult to make a clear
discrimination of the strong lensing, because the magnification is
not a direct observable. One advantage of the halo model developedin this paper is that it allows a fair comparison with the measurement by employing the selection criteria in the measurement for themodel prediction. Based on these considerations, we derived useful,general dependences of the magnification correlation amplitude onlarge magnifications (µ  1), from the universal lensing properties
in the vicinity of critical curves (see Section 4). The intriguing consequence is that, for a correlation function parametrized as µp f ,
the amplitude converges to be finite for p < 1 and otherwise diverges p  1 as the maximum magnification cut-off µmax → ∞,independent of details of the lensing mass distribution. This wasquantitatively verified by the halo model prediction as well as by
the simulations (see Fig. 5). This result therefore implies that themagnification statistics with p  1 is practically advantageous inthat it is insensitive to the selection effect of the magnification cutoff µmax . This is the case for the two-point correlation of size (notarea) fluctuations of distant galaxy images and for the QSO–galaxy
cross-correlation, if the unlensed number counts of QSOs with alimiting magnitude have a slope of s = d ln N(m)/dm < 4/5.
It might be imagined that the non-linear magnification contribution can be suppressed by clipping regions of cluster of galaxies fromsurvey data in order to avoid the uncertainty in the model prediction
and to apply the weak lensing approximation (e.g. see Barber &Taylor 2003). However, this likely adds an artificial selection effectin the analysis and causes a biased cosmological interpretation. In
addition, the lensing projection makes it relatively difficult to correctly identify the cluster region from the reconstructed convergencemap, unless accurate photo-z information or follow-up observations
are available (e.g. White, Van Waerbeke & Mackey 2002). The approach of this paper allows us to treat data more objectively.
There are some effects we have so far ignored in the halo model
calculation. The most important one is the asphericity of the halo
profile in a statistical sense. The aspherical effect could lead to substantial enhancement of the magnification correlation amplitude,
because it is known that an area enclosed by the critical curve islargely increased by ellipticity of the lensing mass distribution, thusincreasing the cross-section of high magnifications to the correlation evaluation. For example, the number of strong lensing arcs due
to clusters of galaxies is amplified by an order of magnitude if we
consider an elliptical lens model instead of an axially symmetricprofile (e.g. Meneghetti, Bartelmann & Moscardini 2003; Oguri,
Lee & Suto 2003). For the same reason, substructures within a halo
could have strong impact on the magnification correlation, as theynaturally emerge in the CDM simulations. Hence, it is of great interest to investigate the effect on the magnification statistics withhigher-resolution simulations.
AC K N OW L E D G M E N T S
We are grateful to Bhuvnesh Jain for collaborative work, valuable
discussions and a careful reading of the manuscript. TH is grateful for the warm hospitality received at University of Pennsylvania,
where part of this work was carried out. We also thank M. Bartelmann and M. Oguri for valuable discussions. TH also acknowledges
supports from the Japan Society for Promotion of Science (JSPS)
Research Fellowships. Numerical computations presented in thispaper were partly carried out at the Astronomical Data Analysis
Centre (ADAC) of the National Astronomical Observatory, Japan

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and at the computing centre of the Max-Planck Society in Garching.
The N-body simulations used in this paper were carried out by the
Virgo Supercomputing Consortium.
REFERENCES
Barber A. J., Taylor A. N., 2003, MNRAS, 344, 789
Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15
Bartelmann M., 1995, A&A, 298, 661
Bartelmann M., 1996, A&A, 313, 697
Bartelmann M., Narayan R., 1995, ApJ, 451, 60
Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291
Benı́tez N., Martı́nez-González E., 1997, ApJ, 477, 27
Benı́tez N., Sanz J. L., Martı́nez-González E., 2001, MNRAS, 320, 241
Blandford R. D., Narayan R., 1986, ApJ, 310, 568
Blandford R. D., Saust A. B., Brainerd T. G., Villumsen J. V., 1991, MNRAS,
251, 600
Broadhurst T. J., Taylor A. N., Peacock J. A., 1995, ApJ, 438, 49u
Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., Kravtsov A. V., Klypin
A. A., Primack J. R., Dekel A., 2001, MNRAS, 321, 559
Cooray A., Sheth R., 2002, Phys. Rep., 372, 1
Dodelson S. et al., 2002, ApJ, 572, 140
Dolag K., Bartelmann M., 1997, MNRAS, 291, 446
Gaztañaga E., 2003, ApJ, 589, 82
Guzik J., Seljak U., 2002, MNRAS, 335, 311
Hamana T., 2001, MNRAS, 326, 326
Hamana T., Mellier Y., 2001, MNRAS, 327, 169
Hamana T., Martel H., Futamase T., 2000, ApJ, 529, 56
Hamana T. et al., 2003, ApJ, in press (astro-ph/0210450)
Hattori M., Kneib J.-P., Makino N., 1999, Prog. Theor. Phys. Suppl., 133, 1
Hu W., Kravtsov A., 2003, ApJ, 584, 702
Jain B., 2002, ApJ, 580, L3
Jain B., Scranton R., Sheth R., 2003, MNRAS, 345, 62
Jain B., Seljak U., White S. D. M., 2000, ApJ, 530, 547
Jarvis M., Bernstein G., Jain B., Fischer P., Smith D., Tyson, J. A., Wittman
D., 2003, AJ, 125, 1014
Kauffmann G., Colberg J. M., Diaferio A., White S. D. M, 1999, MNRAS,
307, 529
Kaiser N., 1992, ApJ, 388, 272

Mao S., 1992, ApJ, 389, 63
Mellier Y., 1999, ARA&A, 37, 127Ménard B., Bartelmann M., 2002, A&A, 386, 784
Ménard B., Hamana T., Bartelmann M., Yoshida N., 2003, A&A, 403,
817
Meneghetti M., Bartelmann M., Moscardini L., 2003, MNRAS, 340, 105
Miralda-Escude J., 1991, ApJ, 380, 1
Moessner R., Jain B., 1998, MNRAS, 294, L18
Navarro J., Frenk C., White S. D. M., 1997, ApJ, 490, 493 (NFW)
Oguri M., Lee J., Suto Y., 2003, ApJ, in press (astro-ph/0306102)
Peacock J. A., 1982, MNRAS, 199, 987
Peebles P. J. E., 1980, Large-Scale Structure of the Universe. Princeton Univ.
Press, Princeton, NJ
Sanz J. L., Martinez-González E., Benitez N., 1997, MNRAS, 291, 418
Schneider P., 1987, ApJ, 319, 9
Schneider P., Ehlers J., Falco E. E., 1992, Gravitational Lenses. Heidelberg,
SpringerSeljak U., 2000, MNRAS, 318, 203
Seljak U., Zaldarriaga M., 1996, ApJ, 469, 437
Smith R. E. et al., 2003, MNRAS, 341, 1311
Sheth R. K., Tormen G., 1999, MNRAS, 308, 119
Spergel D. N., Steinhardt P. J., 2000, Phys. Rev. Lett., 84, 3760
Sugiyama N., 1995, ApJS, 100, 281
Takada M., Jain B., 2002, MNRAS, 337, 875 (TJ02)
Takada M., Jain B., 2003a, ApJ, 583, L49 (TJ03a)
Takada M., Jain B., 2003b, MNRAS, 340, 580 (TJ03b)
Takada M., Jain B., 2003c, MNRAS, 344, 857 (TJ03c)
Vale C., White M., 2003, ApJ, 592, 699
Van Waerbeke L., Hamana T., Scoccimarro R., Colombi S., Bernardeau F.,
2001, MNRAS, 322, 918
Vietri M., Ostriker J., 1983, ApJ, 267, 15
White M., 2002, ApJS, 143, 241
White M., Van Waerbeke L., Mackey J., 2002, ApJ, 575, 640
Yoshida N., Sheth R., Diaferio A., 2001, MNRAS, 328, 669
Yoshida N., Springel V., White S. D. M., Tormen G., 2000, ApJ, 544, L87
Zehavi I. et al., 2003, ApJ, submitted (astro-ph/0301280)
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2003 RAS, MNRAS 346, 949–962