A&A 431, 925 (2005)
DOI: 10.1051/00046361:20034217
The threepoint correlation function of cosmic shear
II. Relation to the bispectrum of the projected mass density and generalized
thirdorder aperture measures
P. Schneider
^{1} 
M. Kilbinger^{1}

M. Lombardi^{1,2}
1  Institut f. Astrophysik u. Extr. Forschung, Universität Bonn,
Auf dem Hügel 71, 53121 Bonn, Germany
2 
European Southern Observatory, KarlSchwarzschildStr. 2,
85741 Garching, Germany
Received 21 August 2003 / Accepted 25 September 2004
Abstract
Cosmic shear, the distortion of images of highredshift sources by the
intervening inhomogeneous matter distribution in the Universe, has
become one of the essential tools for observational cosmology since
it was first measured in 2000. Since then, several surveys have been
conducted and analyzed in terms of secondorder shear
statistics. Current surveys are on the verge of providing useful
measurements of thirdorder shear statistics, and ongoing and future
surveys will provide accurate measurements of the shear threepoint
correlation function which contains essential information about the
nonGaussian properties of the cosmic matter distribution.
We study the relation of the threepoint cosmic shear statistics to
the thirdorder statistical properties of the underlying convergence,
expressed in terms of its bispectrum. Explicit relations for the
natural components of the shear threepoint correlation function
(which we defined in an earlier paper) in terms of the bispectrum are
derived. The behavior of the correlation function under parity
transformation is obtained and found to agree with previous
results. We find that in contrast to the twopoint shear correlation
function, the threepoint function at a given angular scale
is not affected by power in the bispectrum on much larger scales. These
relations are then inverted to obtain the bispectrum in terms of the
threepoint shear correlator; two different expressions, corresponding
to different natural components of the shear correlator, are obtained
and can be used to separate E and Bmode shear contributions. These
relations allow us to explicitly show that correlations containing an
odd power of Bmode shear vanish for paritysymmetric
fields. Generalizing a recent result by Jarvis et al., we derive
expressions for the thirdorder aperture measures, employing multiple
angular scales, in terms of the (natural components of the)
threepoint shear correlator and show that they contain essentially
all the information about the underlying bispectrum. We discuss the
many useful features these (generalized) aperture measures have that
make them convenient for future analyses of the skewness of the cosmic
shear field (and any other polar field, such as the polarization of
the Cosmic Microwave Background).
Key words: cosmology: largescale
structure of the Universe
1 Introduction
Recent surveys have measured secondorder cosmic shear statistics with
high accuracy, owing to the large sky area covered, and thus the large
number of faint galaxy images (e.g., van Waerbeke et al. 2001, 2002;
Jarvis et al. 2003a; Hoekstra et al. 2002). With surveys of this
size, it now becomes feasible to obtain higherorder cosmic shear
statistics which probe the nonGaussian features of the cosmic shear
field. These higherorder statistics are particularly useful in
breaking neardegeneracies in cosmological parameters which are
present at the level of secondorder statistics. Bernardeau et al. (1997), van Waerbeke et al. (1999) and others pointed out that the
skewness of the convergence underlying the cosmic shear field can
break the degeneracy between the density parameter
and the normalisation of the matter power spectrum, expressed in terms
of the rms density fluctuations
on a scale of
.
However, the convergence cannot be observed
directly, but needs to be inferred from the observed galaxy image
ellipticities which yield an estimate of the local shear. The
dispersion of the shear in a (circular) aperture, frequently used to
quantify secondorder shear statistics, cannot be generalized to
a thirdorder statistics. Schneider
et al. (1998, hereafter SvWJK) have defined an alternative cosmic
shear measure, the aperture mass, which is a scalar quantity that can
be directly obtained from the shear, and therefore is particularly
suited to define higherorder statistics.
Recently, interest in higherorder cosmic shear statistics has been
revived. The threepoint correlation function (3PCF henceforth) of the shear
contains all the information on the thirdorder statistical properties of the
shear field, and therefore is of prime interest. In addition, it can be
obtained directly from the observed image ellipticities and, in contrast to
the aperture mass statistics, is insensitive to holes and gaps in the data
field. However, the shear 3PCF is a function with
components (since each shear has two independent components) and 3
variables (e.g., the sides of the triangle formed by the three points)
and therefore difficult to handle. Bernardeau et al. (2003) defined a
specific integral over the 3PCF and applied that to the VIRMOSDESCART survey
in Bernardeau et al. (2002) to obtain the first detection of a nonzero
thirdorder cosmic shear signal. Using the same observational data, Pen et al. (2003) calculated the skewness of the aperture mass, where the latter has
been obtained from integrating the shear 3PCF. Jarvis et al. (2003b, JBJ
hereafter) obtained an alternative expression for the aperture mass skewness
in terms of the shear 3PCF and applied this to the CTIO cosmic shear survey,
finding a signal at about the 2.3 level.
Following a different approach, the 3PCF was considered directly in a
number of recent papers. Schneider & Lombardi (2003, hereafter
Paper I) defined special combinations of the shear 3PCF which we termed
the "natural components'', because they obey simple
transformation laws under coordinate rotations. In particular, we
derived the behavior of the 3PCF under parity transformations, and
showed that all eight components are expected to be nonzero for a
general triangle configuration. Zaldarriaga & Scoccimarro (2003) and
Takada & Jain (2003a) obtained analytic approximations and numerical
results, using raytracing simulations, for the 3PCF. Takada & Jain
(2003b) provide an extensive study of the expectation for the shear 3PCF
in terms of the halo model of the largescale structure, and they
verified the accuracy of their analytical results with numerical
simulations. Schneider (2003) investigated the transformation
properties of a general 3PCF of a polar under parity transformations
and showed that the expectation value of any quantity containing an
odd power of Bmodes vanishes for parityinvariant shear fields.
In this paper, we first consider the relation between the shear 3PCF and the
bispectrum of the underlying convergence (or projected density)
field. If the shear field is derivable from a scalar potential (that
is, a pure Emode field), as expected for cosmic shear in the absence
of intrinsic galaxy alignments and systematics in the observing
process, the bispectrum of the convergence fully encodes the
thirdorder information of the random field^{}. In this first part of the paper, we
therefore generalize the relations between the power spectrum of the
convergence and the various secondorder shear statistics (derived in
Crittenden et al. 2002; Schneider et al. 2002) to thirdorder shear
statistics. After some
preliminaries in Sect. 2, we derive the shear 3PCF in terms of the
bispectrum of the convergence. From these explicit relations, general
transformation laws of the 3PCF can be directly seen; for example, the
behavior of the 3PCF under parity inversion as studied before in Paper I and
in Schneider (2003) can be explicitly
verified, as will be shown in Sect. 4. In Sect. 5 we invert these
relations, i.e., we express the bispectrum in terms of the 3PCF of the
shear. We obtain two formally different expressions for the bispectrum
which must,
however, be identical in the case of a pure Emode shear field.
In the second part of this paper (Sect. 6), we consider the
thirdorder aperture mass statistics as a particularly convenient
integral over the shear 3PCF; in fact, this part of the paper will
quite likely be most relevant for future studies of higherorder
cosmic shear statistics. We first express
in terms of the bispectrum and then replace
the bispectrum in terms of the 3PCF. This procedure
yields the same result for
in terms of
the shear 3PCF as derived by JBJ. We then argue that the thirdorder
aperture measures contain only part of the information about the
bispectrum of the underlying convergence, and generalize the aperture
measures to the case of three different scale lengths. We show that
this generalization allows us to obtain essentially the full
information about the bispectrum. These generalized aperture measures
are then expressed in terms of the shear 3PCF. Section 7
summarizes and discusses our results.
It must be stressed that all our results are valid for other random
fields which share the properties of that of cosmic shear: A
homogeneous, isotropic, paritysymmetric random field of a polar. The
most obvious example of such a field in the cosmological context,
apart from cosmic shear, is the polarization field of the Cosmic
Microwave Background (e.g., Zaldarriaga et al. 1997)
2 Preliminaries
In the first part of this paper (through Sect. 5) we shall
consider a shear field
which is caused by an underlying projected
density (or convergence) field ,
as is expected for a shear field
produced by light propagation in an inhomogeneous Universe (e.g., Blandford et al. 1991; MiraldaEscudé 1991; Kaiser 1992; see also the reviews by
Mellier 1999 and Bartelmann & Schneider 2001). The relation between the shear
(expressed throughout this paper as a complex number) and the convergence is
most simply given in Fourier space,

(1) 
where
is the polar angle of
,
and
is the Fourier
transform variable of the angular position vector on the sky.
In Paper I we considered the 3PCF of the shear. Since the shear is a
twocomponent quantity, the 3PCF has 8 independent components. Since one
cannot form a scalar from the product of three shears alone, one needs to
project the shear with respect to some reference directions. The three points
at which the shear is considered form a triangle, and one can project the
shear along directions attached to such a triangle, i.e., which rotate with
the triangle in coordinate rotations. We have considered a number of such
obvious projections, namely with respect to the directions of the vertices
towards one of the centers of a triangle. Let
be the polar angle of
the vector connecting the point
with the chosen center, then the
Cartesian components
of the shear
are used to define the tangential and cross components of the
shear relative to the chosen direction ,

(2) 
If the reference directions are changed, the tangential and cross
components of the
shear will change correspondingly. In particular, defining the 3PCF of the
shear components, they will change if a different center of the triangle is
chosen. In Paper I we defined four complex "natural components'' of the
shear 3PCF which show a simple behavior under such transformations; they have
been termed
,
.
The
,
i=1,2,3 can be obtained from one another by permutations of the arguments x_{i}, which represent the sides of the triangle.

Figure 1:
Definitions of the geometry of a triangle. The
are the vertices of the triangle, the
the
corresponding sides,
are the orientations of the sides relative to the
positive x_{1}direction, the
are the interior angles of
the triangle. The figure also shows the orthocenter H, i.e. the intersection of the heights of the triangle. 
Open with DEXTER 
We specify our geometry in the same way as in Paper I (see Fig. 1):
let
be three points at which the shear is considered. The
connecting vectors are
,
,
and
.
We denote the polar angle of the vector
by ,
and the interior angle of the triangle at the point
by .
Furthermore, we assume that the triangle is oriented such that
,
where for two twodimensional vectors
and
we defined
.
Hence, the points
are
ordered counterclockwise around the triangle.
In the first part of this paper, we shall consider the projection of
the shear relative to the orthocenter of the triangle. As the vector
connecting the point
and the orthocenter is perpendicular to
the vector
(see Fig. 1), one has
in
this case, so that

(3) 
The shear 3PCF depends linearly on the 3PCF of the convergence, or
equivalently on its Fourier transform, the bispectrum.
The bispectrum of the surface mass density is defined as (see, e.g.,
van Waerbeke et al. 1999)

(4) 
hence, it is nonzero only for closed triangles in space. This follows from the assumed statistical homogeneity of the random field .
Furthermore, if
is an isotropic random field, the function
depends only on
,
,
and the angle
enclosed by
and
.
We shall therefore write
.
If, in addition, the
statistical properties of the field
are invariant under parity
transformation (as we shall assume throughout), then b is an even function
of ,
or, equivalently, b is invariant against exchanging
and ,

(5) 
3 The shear threepoint correlation function in terms of
the bispectrum
In this section, we will derive the shear 3PCF in terms of the bispectrum Bof the convergence. As it turns out, the calculations are fairly cumbersome,
owing to the rich mathematical structure of the 3PCF with its three
arguments, compared to the 2PCF which has only one argument (and for which the
direction along which the shear components are measured is uniquely given by
the connecting vector between any pair of points).
3.1 The case of
The natural component
of the shear 3PCF measured
relative to the orthocenter reads
where we made use of the relation (1) between the Fourier
transforms of the shear and the convergence, and the definition (3) of the shear components relative to the orthocenter.
Inserting the bispectrum (4) into Eq. (6), performing
for each of the resulting three terms the integration over the
vector which is not in the argument of the B function, by making
use of the delta"function'' in (4), and using the relations
between the corner points
and the side vectors
,
one
obtains after renaming the dummy integration variables
The angle
occurring in (7)
is the polar angle of the vector
,
so that
We next
rename the angles in the following way:

(8) 
so that
is the angle between
and
,
as previously
defined, and
is the angle between the direction of
and
the mean of the directions of
and .
Since B is
independent of ,
a further integration can be carried out in
Eq. (7), using
.
Owing to the symmetric
form of the three terms occurring, only one of the three terms has to be
calculated explicitly; we shall consider the first term in the following. From
the foregoing equations one finds that

(9) 
Next, we consider the argument of the exponential,
where we have defined
and used the
fact that
.
Therefore, we can write

(11) 
from which one finds, after expanding the trigonometric functions in
Eqs. (10) and (11),







(12) 
Finally, we consider the sums over the angles that occur in
(7),

(13) 
We can now perform the integration of the first term in
Eq. (7) as follows:
where
is the Bessel function of the first kind. Therefore,
Eq. (7) becomes
where the A_{i} and
are obtained from Eq. (12) by
cyclic permutations of the
x_{1},x_{2},x_{3}.
3.2 The case of
Next, we calculate the natural component
where we made use of the fact that
,
since
is
a real field. Next, we change the integration variable
;
as a consequence,
,
but this
does not change the exponential in Eq. (16). Inserting the
bispectrum in the form Eq. (4), and appropriately renaming
the dummy integration variables, one finds
The further calculation proceeds in the same way as in the case of
.
Specifically, we employ the change of angular
integration variables given in Eq. (8), evaluate the
three exponentials containing products of the form
using Eqs. (10), (11), and their
analogous expressions obtained by cyclic permutations of the x_{i},
and calculating the angular sums in the exponentials. The final result
reads
The expressions for the other two natural components
and
of the shear 3PCF, which are defined in analogy to
Eq. (16) by placing the complex conjugation of the shear at
point
and
,
respectively, are obtained from
Eq. (18) by applying the transformation laws given in Paper
I, i.e., even permutations of the arguments.
The resulting expressions (15) and (18) are not only
relatively complicated, but their numerical evaluation also is quite
cumbersome. Recalling that the relation between the 2PCF of the shear and the
power spectrum
of the convergence involves a convolution
integral over a Bessel function, one should perhaps not be too surprised that
in the case of thirdorder statistics there are three such oscillating factors
in the transformation between the shear 3PCF and the bispectrum. In a
future work, we will investigate numerical procedures with which the
integration can be carried out accurately; first attempts, using
Gaussian quadrature for the two integrations and an equidistant
grid for the integration already yielded satisfactory
results. Hence, despite the apparent complexity, the foregoing
equations can be applied in practice.
4 Transformation laws
In PaperI the behavior of the natural components under a change of
the order
of the arguments was discussed, using simple geometrical arguments. We shall
now consider this behavior explicitly, using the expressions (15)
and (18). First, consider
.
Taking an even permutation of the arguments of
just
changes the order of the terms in the integral of Eq. (15) and
therefore leaves
unchanged. Taking an odd permutation
of the arguments means that two of the arguments are interchanged,
e.g., x_{1} and x_{2}. Interchanging x_{1} and x_{2} corresponds to an
interchange of
and .
Using the property Eq. (5), one can also interchange
and .
From Eq. (9) one sees that these changes imply that
.
Furthermore, from Eq. (12), one
sees that these changes lead to
,
and
.
Together this implies that these transformations lead to a
complex conjugation of the first term in Eq. (15). From the
expressions for A_{l} and
obtained from Eq. (12)
by cyclic permutations of the x_{k}, one finds that the above
interchanges of x_{1} and x_{2} leads to
,
,
,
.
This then implies
that the second term in Eq. (15) becomes the complex conjugate
of the third term, and vice versa. Taken together, we see that an odd
permutation of the arguments changes
,
as already argued from parity considerations in
PaperI.
Note that an odd permutation of the arguments in geometric terms means that
the orientation of the triangle is reversed. We shall now show that this is
equivalent to replacing all
by .
The motivation for this
observation comes from the fact that for a triangle with odd orientation, the
relations (see Eq. (1) of Paper I) between the orientations
of the
and the interior angles
formally yield
(modulo ), whereas for a triangle with even
orientation,
(modulo ). If we apply this
transformation,
,
we can change the integration variable
in Eq. (15), noting from Eq. (5) that
is unaffected by this change. These two changes
together then imply that
,
,
and
(see Eqs. (9) and (12),
respectively). Hence, all three terms of Eq. (15) are transformed to
their complex conjugates, as was claimed above. Note that this transformation
behavior directly implies that
is real if two of its arguments
are equal.
Next one can consider the transformation behavior of
.
Cyclic permutations of the arguments transform
into
and
,
yielding the
transformation behavior derived in PaperI. Interchanging x_{2} and x_{3} (and thus
and )
should yield the complex conjugate of
.
Again, we
interchange
and ,
which then yields
,
,
,
and so the second
term in Eq. (18) is complex conjugated. Furthermore, these
transformations yield
,
,
,
,
which shows that the first
term in Eq. (18) becomes the complex conjugate of the third,
and vice versa, so that
,
as was to be
shown. This transformation implies that
is real if the
last two arguments are equal.
Equivalently, we can also let
,
and change the
integration variable
.
This implies
,
,
,
and each term in
Eq. (18) is transformed into its complex conjugate.
5 Bispectrum in terms of the 3PCF
Recall the situation for the twopoint correlation of the shear: there, the
relation between the correlation functions and the power spectrum of the
projected density fluctuations can be inverted (e.g., Schneider et al. 2002;
hereafter SvWM), and thus the power spectrum can be expressed in terms of the
twopoint correlation function. It will be shown here that in analogy, the
bispectrum
can be expressed in terms of the 3PCF.
5.1 Bispectrum in terms of
From (1) one finds that
where in the second step we used Eq. (3). We now split the
righthand side of the foregoing equation into three identical terms,
each of which is thus one third of the above expression, and
substitute
,
in the
first of these, and similar substitutions, obtained by cyclic
permutations of the
and
in the other two
terms. This then yields

(20) 
where we have used the notation
for
the 3PCF
,
with
,
if
,
and
if
.
Hence, if expressed in terms of the arguments
,
the information about the orientation of the
three points
is included. Another notation to be used
later on is
,
which also includes the
orientation of the three points. The
integration in the
previous equation yields a deltafunction; comparison with
Eq. (4) then results in

(21) 
It is easy to show that Eq. (21) is compatible with
Eq. (7), since
where the
are the polar angles of the
,
and
.
The
integrations can be carried
out, yielding delta"functions'': for the first term, e.g., they yield
,
so
that also
.
Together, this yields

(23) 
but since
,
the compatibility of
Eqs. (7) and (21) has been shown.
Next, we want to calculate
from Eq. (21), i.e. taking a further integration. For that
purpose, we write
,
,
where
.
Then,
where we defined
in the second step, and A_{3} and
are given in Eq. (12). Writing the polar angle of
as
,
one finds that
,
where

(25) 
and
is given in Eq. (9). Taken together,
Eq. (21) becomes
It is easy to see that
as given by Eq. (26) is real: since
 see Eq. (5)  and
,
taking
the complex conjugate of Eq. (26) and simultaneously
replacing
,
and the integration variable
yields the same expression as Eq. (26),
.
5.2 Bispectrum in terms of the other
Next we calculate the bispectrum as a function of the other three natural
components of the shear 3PCF, starting from
where we used

(28) 
In complete analogy to Eq. (21), we split the righthand side into
three terms and make appropriate substitutions. From a comparison with
Eq. (4) we then obtain
Here, we wrote the 3PCF as
.
When renaming the integration variables, we have to apply the
transformation rules to the 3PCFs (see Paper I). For the second term, we
perform the substitutions
,
so that
.
The third term is transformed similarly, and we get
Unfortunately, the three terms on the righthand side are
not equal, as was the case for Eq. (20). Therefore, we repeat
the above procedure with
and
.
The two resulting equations are
and
Now, we can sum Eqs. (30)(32), after moving the
phase factors to the lefthand side. Then, we indeed get three
equal terms, therefore
with

(34) 
Equation (33) can be written as a function of
only, again using the transformation properties of the 3PCFs
and
.
We use Eq. (17) and its counterparts for
and
and insert them into Eq. (33). The phase
factors cancel, so that one obtains
with
,
where
is the polar angle of the vector
.
The x_{1} and x_{2}integrals can be performed and yield
functions. This makes the k_{1} and k_{2}integrals trivial,
yielding
and
.
All the phase exponentials add up to give 3 g, canceling the
prefactor in Eq. (35), leaving only
on
the right hand side and thus verifying Eq. (32).
As was the case for Eq. (26), one additional angular integral can be
carried out. With

(36) 
we find for the three terms:

(37) 
In all three cases, we get the integral
.
Finally,
5.3 Comments
After having seen the relations of the 3PCF in terms of the bispectrum, one
is not surprised to find that their inversion derived in this section also is
of considerable complexity. This can again be compared to the
case of secondorder statistics, where the power spectrum can be written in
terms of the correlation function through an integration. The integration
extends over all angular scales (as is also the case here), and so the direct
inversion will always be of limited accuracy since the correlation functions
can only be measured on a finite range of angular scales. In order to see the
range of application of the previous relations, numerical simulations are
probably required.
The foregoing equations also allow us in principle to express the 3PCF
in terms of
,
by using the expression (17) for
and substituting the bispectrum in this
equation by
,
using Eq. (21). Whereas the corresponding
equations can be reduced to a threedimensional integral, they are fairly
complicated; therefore, we shall not reproduce them
here.
Given the measured correlation functions from a cosmic shear survey, there is
no guarantee that the bispectrum estimates from Eqs. (26) and (38) will agree, even if we ignore noise and measurement
errors. The two results will agree only if the shear field is derivable from
an underlying convergence field, i.e., if the shear is a pure Emode
field. Significant differences between the two estimates would then signify
that there is a Bmode contribution to the shear. In the case of
secondorder statistics, the separation between E and Bmodes is most
conveniently done in terms of the aperture statistics (see Crittenden et al. 2002, hereafter CNPT); we shall therefore turn to the aperture measures of the
thirdorder shear statistics in the next section.
At first sight, it may appear surprising that the expression (26)
for the bispectrum is always real, even though we have not constrained
to correspond to a pure Emode field (in fact, we would not
really be able to put this constraint on the 3PCF  compare the 2PCF: only by
combining the two correlation functions
can one separate E from
Bmodes). The only assumption we made was that the shear field is parity
invariant. This can be understood as follows: We can describe a general shear
field by the Fourier transform relation (1) if we formally
replace the convergence by
,
where
gives rise to a pure Emode
shear field, and
corresponds to a pure Bmode shear
(SvWM). Considering the triple correlator of this complex ,
one finds
that its real part consists of terms
and
,
whereas the imaginary part has
contributions
and
.
As shown by Schneider (2003), the latter two terms are strictly zero
for a parityinvariant shear field, so that the fact that Eq. (26)
is real is fully consistent with the vanishing of the imaginary part of the
triple correlator of the complex
 both are due to the assumed
parity invariance. This argument then also implies that the resulting
expression (26) for b contains both E and Bmodes.
One can separate E and Bmodes of the bispectrum by suitably combining the
expressions (26) and (38). As we shall discuss in
Sect. 7, the Emode bispectrum is obtained by

(39) 
6 Aperture statistics
We have seen that it is possible to calculate the 3PCF in terms of the
bispectrum, and in principle also to invert this relation. However,
the resulting integrals are very cumbersome to evaluate numerically,
owing to the various oscillating factors. It therefore would be useful
to find some statistics that can be easily calculated in terms of the
directly measurable 3PCF, but which can also be easily related to the
bispectrum. In their very interesting paper, JBJ
considered the aperture measures, which have been demonstrated to be
very useful in the case of secondorder statistics. The aperture mass
centered on the origin of the coordinate system is defined as

(40) 
where
is a filter function of characteristic radius
,
the filter function

(41) 
is related to
,
and the second equality in Eq. (40) is
true as long as
is a compensated filter,
i.e.
,
as has been shown by Kaiser et al. (1994) and Schneider (1996).
is the shear component tangent
to the center of the aperture, i.e., the origin. Hence,
,
where here and in the following we use the notation that a vector
can also be represented by a complex number
.
Hence,
is nothing but the phase factor
,
where
is the polar angle of
.
The aperture mass as a statistics for cosmic shear was introduced by
SvWJK who showed that the dispersion
of the aperture mass is given as the integral over the
power spectrum of the projected mass density ,
convolved with
a filter function which is the square of the Fourier transform of .
SvWJK derived this filter function for a family of
functions
which have a finite support. These filter
functions turn out to be quite narrow, so that
provides very localized information about the power
spectrum (see also Bartelmann & Schneider 1999). Furthermore, SvWJK
calculated the skewness of
in the frame of
secondorder perturbation theory for the growth of structure. As it
turned out, the resulting equations are quite cumbersome, which is in
part related to the fact that the Fourier transform of the functions
chosen contains a Bessel function.
CNPT suggested an alternative form of the function
.
When we write
,
then the
filter used by CNPT is

(42) 
Hence, this filter function does not have finite support; this is, however,
only a small disadvantage for employing it since it cuts off very quickly for
distances larger than a few .
This disadvantage is more than
compensated by the convenient analytic properties of this filter.
A further advantage of using aperture measures is that
,
as
calculated from the rightmost expression in (40), is sensitive
only to an Emode shear field (see CNPT and SvWM for a discussion of the
E/Bmode decomposition of shear fields). Hence,
defining the complex number

(43) 
vanishes identically for Bmodes, whereas
yields zero for a pure Emode field. Thus, the
aperture measures are ideally suited to separating E and Bmodes of
the shear.
CNPT and SvWM have shown that the dispersions
and
can be expressed as an
integral over the twopoint correlation functions of the shear. Since
the correlation functions are the best measured statistics on real
data (as they are insensitive to the gaps and holes in the data
field), this property allows an easy calculation of the aperture
dispersions from the data. JBJ showed that the thirdorder moments of
the aperture measures can likewise be expressed by the shear 3PCF, and
they derived the corresponding relations explicitly  they are
remarkably simple. The fact that such explicit results can be obtained
is tightly related to the choice of the filter function (42); for a filter function with strictly finite
support, the resulting expressions are very messy (indeed, we have
derived such an expression for the filter function used in SvWJK, but
it is so complicated that it will most likely be useless for any
practical work).
We shall here rederive one of the results from JBJ making
use of the results obtained in Sect. 5; the agreement
of the resulting expression with that of JBJ provides a convenient
check for the correctness of the results in Sect. 5.
In a first step, we
express
in terms of the bispectrum. Using the
first definition in Eq. (40), we find that
When carrying out the
integrations, the Fourier transforms
are obtained. Inserting the bispectrum in the form Eq. (4) and integrating out the corresponding delta function, one
obtains three identical terms. With
one finds
We shall discuss this result in the next subsection; here, we want to use
Eq. (45) and obtain an explicit equation for
in
terms of the 3PCF of the shear. For better comparison with JBJ,
we shall slightly change our notation for the 3PCF. Up to now we have
labeled the sides of the triangle formed by the three points
by the
vectors
as defined in Sect. 2. The corresponding
natural components of the 3PCF were then denoted by
.
We shall now define
the three points
in the form
,
,
and then define

(47) 
where the "x'' denotes an arbitrary projection of the shear components,
i.e. relative to an arbitrary choice of reference directions. The relation
between the
and the
follows simply from the
definitions of the separation vectors between the points
,
which is
,
,
so that

(48) 
and analogously for the other components of the 3PCF.
Now, from combining Eqs. (21) with (45) and using our new
notation for the 3PCF, one finds

(49) 
where the subscript "cart'' denotes the Cartesian components of the 3PCF.
Inserting the Fourier transforms of u from (42) and using
(where we used, as before, the notation
),
(49) becomes
The integrations can now be carried out, by noting that the
exponential
is just a quadratic function of the integration variables, and it is
multiplied by a polynomial. The straightforward, but tedious calculation has
been carried out with Mathematica (Wolfram 1999); the result of the
integration then depends on the
.
Substituting the
in
favor of the vectors

(51) 
which are the vectors connecting the center of mass of the triangle with its
three corners, one can put the result in the form

(52) 
Employing now the transformation laws of the natural components of the 3PCF as
derived in Paper I, one sees that the squares of the
complex conjugates of the q_{i} can be used to obtain the 3PCF with the shear
projected along the direction towards the center of mass of the triangle,
i.e.,

(53) 
After this projection of the 3PCF, the integrand depends only on the absolute
values of the
and the angle
between them. By carrying out one
more integration, one finally obtains

(54) 
with

(55) 
and thus

(56) 
The result (54) agrees with Eq. (44) of JBJ, after the
projection of the 3PCF has been accounted for. It should be noted that in the
case considered here, where the 3PCF was explicitly obtained in terms of the
bispectrum of the convergence which, owing to parity invariance, is real
(cf. the discussion in Sect. 5.3), the expression
(46) is real, and hence Eq. (54) also is real in this
case. This can be seen explicitly, since the value of
at
is just the complex conjugate
one of that at .
We like to point out that this derivation has shown two interesting
aspects: first, the natural component
of the shear 3PCF
arises naturally in this context, confirming the hypothesis of Paper I that this combination of components of the correlation functions is
indeed useful. Second, the derivation shows that the projection of the
shear onto the centroid is the most convenient projection in this
particular application.
6.2 Generalization: Thirdorder aperture statistics with
different filter radii
How important is the thirdorder aperture statistics for investigating the
thirdorder statistical properties of the cosmic shear? In order to discuss
this question, we shall first consider the analogous situation for the
secondorder statistics. There, as mentioned before, the aperture mass
dispersion is a filtered version of the power spectrum
of the
underlying convergence; for the function
considered here, one has

(57) 
hence, the filter function relating
and
is very narrow, and unless the power spectrum
exhibits sharp features, the function
contains
basically all the information available for secondorder shear statistics (not
quite  see below).
The analogous equation to Eq. (57) for thirdorder statistics is
given in Eq. (46). The function
is very narrowly peaked at
around
,
and there is one factor of
for each of the
three sides of a triangle in space. This implies that in the
integration of Eq. (46) the bispectrum is probed only in regions of
space where
.
Thus,
probes
the bispectrum essentially only for equilateral triangles in Fourier
space. For this reason, the function
cannot carry
the full information of the bispectrum; it merely yields part of this
information.
On the other hand, (46) immediately suggests how to improve
this situation: if we define the aperture mass statistics with three different
filter radii ,
we can probe the bispectrum at wavevectors whose
lengths are
,
and by covering a wide range of
,
one can essentially probe the bispectrum over the full
space. Indeed,
which illustrates what was said above. Thus, this thirdorder statistics is
expected to be as important for the thirdorder shear statistics as is the
aperture mass dispersion for secondorder shear statistics. The fact that we
do not gain additional information by considering different filter scales for
the secondorder
statistics follows from the fact that

(59) 
Whereas the fact that the mixed correlator can be expressed exactly in terms
of the dispersion at an average angle depends on the special filter function
considered here, it nevertheless shows that one does not gain additional
information when considering the covariance of
.
We shall now calculate the triple correlator of
for three
different filter radii in terms of the shear 3PCF, essentially using the same
method as JBJ. For that, we first calculate the
thirdorder statistics of the complex aperture measure M, as defined in
Eq. (43):
where the
are the polar angles of the vectors
.
Writing, as
before,
,
,
replacing the
phase factors by
,
and inserting the definitions of the ,
one obtains
where we set for ease of notation the dummy variable
.
The Yintegration is again over the exponential of a secondorder polynomial
in the integration variable, times a polynomial, and thus straightforward to
integrate, but tedious. Employing Mathematica does most of the job, though
its output needed to be further simplified. The result is

(62) 
where

(63) 

(64) 

(65) 
The choice of the various quantities defined above was made such that S,
Z, and the f_{i} are dimensionless, and that they become very simple
if all
are equal. Consider this special case next, i.e., let
.
Then,
,
S=1,
f_{1}=f_{2}=f_{3}=1, and
.
Thus, we recover the result Eq. (54) in this case which was shown
above to agree with the result from JBJ. The difference
between Eqs. (54) and (62) is that the former has been
derived in this paper from the bispectrum of the convergence, and therefore is
strictly real, whereas Eq. (62) has been calculated directly in
terms of the shear 3PCF and thus applies to arbitrary shear fields, containing
both E and Bmodes. For reference, we explicitly give the combinations of the
appearing in the f_{i} above,
One expects that
is symmetric with respect to any
permutation of its arguments. Indeed, one can show explicitly that Eq. (62) is symmetric with respect to interchanging and .
Performing this interchange, changing the variables
of integration as
,
,
,
and
making use of the fact that
,
one finds that these transformations lead to
,
,
,
and Z is unchanged. To show the symmetry
with respect to even permutations of the arguments, one needs to employ
the symmetry of
,
and then use either X_{1} or X_{2} as the reference point in the
derivation. This then leads to a cyclic permutation of the q_{i} and
the f_{i}, and thus leaves Eq. (62) invariant.
In Fig. 2 we show the latter part of the integrand in
Eq. (62) for the case of three equal apertures
(
)
and for different aperture sizes. Its
zeros, if any, are lines of constant y_{2}/y_{1},
because the function only depends on the ratio of y_{2} and y_{1}.

Figure 2:
Contours of the integration function
 see (62)  as a
function of y_{1} and y_{2} for fixed
( upper row:
,
lower row:
).
The leftmost of the three columns represents the case where all
three aperture radii are equal. The function scales with the aperture
radius. Note that the imaginary part vanishes here because of symmetry.
The two right columns show the real and imaginary part of the
integrand for three different filter radii.
The contour lines are logarithmically spaced with a factor of 5
between successive lines,
starting with 10^{10}. Dashed lines correspond to negative values. 
Open with DEXTER 
We next consider the combination of aperture measures
where the semicolon in the arguments of
indicates that
this expression is symmetric with respect to interchanging the first
two arguments, but not the third one, of course.
Using the same conventions for labeling the vertices
of the
triangle as before, we obtain
After performing the Yintegration and a few manipulations to express the
in terms of the
,
we obtain
where we have defined

(69) 
and the f_{i} are as before. This form of the equation is easily compared with
the result obtained by JBJ, by setting
,
so that
,
S=1,
f_{1}=f_{2}=f_{3}=g_{3}=1, and
.
This then reproduces
their Eq. (49), except for a different labeling of the q_{i} (we considered
the complex conjugate shear at the point
,
whereas JBJ did
this at
).

Figure 3:
Contours of the latter part of the integrand in
Eq. (71). The plotted function
is
times the term in square brackets. The upper and
lower row correspond to fixed values of
and respectively. In the left two columns, the real and imaginary part of
the function is shown for the three aperture radii being equal. The
right two panels correspond to three different radii. The contours are
the same as in Fig. 2. 
Open with DEXTER 
We now employ again the relation between the natural components of the shear
3PCF in the Cartesian reference frame and those measured relative to the
center of mass of the triangle (see Paper I),

(70) 
and make the corresponding replacements in Eq. (68), after which one
more angular integration can be carried out, to obtain our final result
which generalizes the result of JBJ for unequal aperture radii. The
proof that this last expression is symmetric with respect to
interchanging
and
is the same as the one given
above.
See Fig. 3 for an exemplary plot of the latter part of the integrand.
The product of the four q_{i}'s can be written as follows,

(72) 
From the two
complex triple correlators
and
,
we can now
calculate the four real thirdorder aperture statistics, in analogy to
what was done in JBJ,
with the same notational convention as used before, e.g.,
,
which is symmetric in the last two arguments, as indicated by the
semicolon. These four expressions have very different physical
interpretations. A significant nonzero value of
indicates that the Emode of the shear field corresponds to a
convergence field
which has significant skewness. This is the
signal one wants to measure in future cosmic shear surveys, and this
term contains the information about the underlying cosmic density
field, and thus about cosmology. A significant nonzero value of
indicates the presence of a Bmode in the
shear field which is correlated with the Emode. Although lensing can
generate such a term with small amplitude, by higherorder lensing
effects (caused by source clustering, violation of the Born
approximation in studying light propagation in the Universe, or
multiple light deflections  see SvWJK for a discussion of these
latter effects), these are probably too small to be
detectable. Therefore, a detection of a
most likely will indicate the presence of a "shear'' not coming from
lensing, but from, e.g., intrinsic alignment of the galaxies (see,
e.g., Catelan et al. 2000; Heavens et al. 2000; Crittenden et al. 2001; Croft & Metzler 2001; Jing 2002). A significant nonzero value
of
indicates that the shear field violates parity
invariance, as a Bmode shear cannot have odd moments if it is
paritysymmetric (Schneider 2003). Finally, a significant nonzero
value of
indicates a parity invariance
violation which is correlated with the Emode shear field. Neither of
these two latter terms can be explained by cosmic effects which are
expected to by parityinvariant, but either indicates an underestimate
of the statistical errors (coming from the intrinsic ellipticity
distribution of the sources and from cosmic variance), or the presence
of instrumental systematics or artifacts from data reduction (cf. the
analogous situation for secondorder statistics, where a nonzero
value of
would indicate significant
systematics).
As stated above, measuring the thirdorder aperture statistics
(through measuring the shear 3PCF and then using the foregoing
relations) yields essentially all the information about the
bispectrum, provided the latter has no sharp features in
space. The analogous statement for the secondorder statistics
is not really true: if one considers a cosmic shear survey consisting
of several unrelated fields of size
each, one can calculate the
aperture dispersion from the shear 2PCF for scales, say,
.
However, the cosmic shear field contains
information about the power spectrum of the convergence from all
scales; in particular, due to the fact that the shear 2PCF is obtained
from the power spectrum through a filter function which tends to
constant for
,
it contains information over the
integrated power on large scales. Hence, the secondorder aperture
statistics do not recover the full information about the power
spectrum contained in the shear 2PCF for a survey of a given size. In
order to make better use of the shear data, one should take into
account a shear measure which contains the largescale power, such as
the tophat shear dispersion on an angular scale comparable to the
size of the observed fields, say at
,
which can also be
obtained in terms of the shear 2PCF (CNPT, SvWM). An analogous
situation does not exist for the thirdorder statistics. This can be
understood intuitively in the following way: consider again the survey
geometry mentioned above, and assume that to each of the independent
fields a constant shear is added, corresponding to very largescale
power and/or power in the bispectrum. The aperture measures will be
unable to measure this constant shear, whereas the shear 2PCF will be
sensitive to it, as will be the tophat shear dispersion. However,
since one cannot form a thirdorder shear statistics which contains
the shear only, i.e., without reference directions (such as the
direction to the centers of triangles), such a constant shear is
expected to leave no trace on the shear 3PCF. This can be seen
geometrically as follows: consider a triangle of points in a constant
shear field. Rotation of this triangle by 90 degrees changes the sign
of all shear components, and thus the triple product changes sign, for
which reason a constant shear yields no shear 3PCF. This can also be
seen algebraically from Eqs. (15) and (18): the
occurrence of the Bessel functions, which behave like
and ,
respectively, for small
(at fixed x_{i}) removes all
largescale contributions of the bispectrum in the 3PCF. This fact
suggests that indeed the thirdorder aperture measures recover
essentially all information about the bispectrum which is present in
the shear field.
One might argue that the skewness of the convergence field, tophat
weighted in a circular aperture, is sensitive to long wavelength
modes, and so thirdorder statistics on small scales knows about large
scales. This is true, and may sound like a contradiction to what has
been said above. Looking at the secondorder statistics first,
is sensitive to the power spectrum on all
scales
,
and it can be expressed by the 2PCF
on angular scales
(CNPT, SvWM). This is due to
the fact that the 2PCF of
is the same as that of
.
Indeed, if one expresses
in terms of
,
the resulting convolution kernel has infinite support; hence,
cannot be expressed through
over a finite
range, because
is not sensitive to the power spectrum on
large scales.
Something analogous happens for the threepoint statistics.
Whereas one can express
in terms of the shear 3PCF
(this in fact is easily done, e.g. by first expressing
in
terms of the bispectrum, and then replacing the bispectrum in terms of
the shear 3PCF, using the relations in Sect.5), the
integration range is infinite. One cannot calculate
from the shear over a finite region  in fact, the masssheet
degeneracy does prevent this. Only on very large fields, where the
mean of
can be set to zero, can one in principle measure
,
and that means, one needs information from much
larger scales than the size of the aperture.
7 Summary and discussion
In this paper we have considered the relation between the 3PCF of the
cosmic shear and the bispectrum of the underlying convergence
field. Explicit expressions for the (natural components of the) shear
3PCF in terms of the bispectrum have been derived. These expressions
are fairly complicated, and their explicit numerical evaluation
nontrivial. The transformation properties of the 3PCF under parity
reversal can be directly studied using these explicit relations and
confirm those derived in Paper I by geometrical reasoning. We have
then inverted these relations, i.e., derived the bispectrum in terms
of the shear 3PCF. Two different expressions were obtained,
corresponding to the two types of natural 3PCF components: one the one
hand
,
and
,
i=1,2,3 on the other
hand. If the shear is due to an underlying convergence field, these
two expressions should yield the same result for the bispectrum; in
general, however, if a Bmode contribution is present, these two
results will differ. Drawing the analogy to the E/Bmode decomposition
for the aperture measures in Sect. 6.2, we have conjectured
a linear combination of the two expressions for the bispectrum which
yields the Emode only. The orthogonal linear combination then yields
the crossbispectrum of the Emode with the square of the Bmode
shear. The fact that the bispectrum is real, provided the 3PCF obeys
parity invariance, reaffirms the result of Schneider (2003) that for a
paritysymmetric field, all statistics with an odd power of Bmodes
have to vanish.
We have then turned to the aperture statistics, using the filter function that
was suggested by CNPT and also used by JBJ. As a first step we have used the
previously derived expressions for the bispectrum in terms of the 3PCF to
rederive one of the results in JBJ. Then, by considering the thirdorder
aperture statistics in terms of the underlying bispectrum we have argued that
the thirdorder aperture statistics with a single filter radius probes the
bispectrum only along a onedimensional cut through its threedimensional
range of definition, namely that of equilateral triangles in
space. Generalizing the aperture statistics to three different filter
radii, the full range of the bispectrum can be probed, and, in analogy to JBJ,
we have derived explicit equations for the generalized thirdorder aperture
statistics in terms of the directly measurable shear 3PCF. We showed
that using different filter radii did not yield additional information
in the case of
secondorder statistics.
The filter function used in the definition of the aperture measures was that
suggested by CNPT. Whereas it does not strictly have finite support, this
disadvantage compared to the filter function defined in SvWJK is outweighed
by the convenient algebraic properties it has; these enabled the explicit
derivation of fairly simple expressions.
Let us summarize the features of the aperture statistics which render
them so useful as a quantity for characterizing cosmic shear (and other polar
fields):

The aperture measures can be directly calculated in terms of the shear
correlation functions. It is the latter that can be measured best from
a real cosmic shear survey, as they are not affected by the geometry
of the survey and holes and gaps in the data field. The expressions of
the aperture measures in terms of the shear correlation function are
easy to evaluate by simple sums over the bins for which the
correlation functions have been measured.

The aperture measures provide very localized information about the
underlying power spectrum (in the case of secondorder statistics) and
the bispectrum (for thirdorder statistics) and therefore contain
essentially the full information about the properties of the
underlying convergence field, unless its power in Fourier space has
sharp features (which is not expected for a cosmological mass
distribution, since there is no sharply defined characteristic length
scale).

One can easily calculate the aperture measures in terms of the power
spectrum and the bispectrum, and hence their expected dependence on
the cosmological parameters can be derived and compared to the
measurements. Whereas the aperture measures are just one particular
way to form integral measures of the shear correlation functions  a
different integral measure was defined by Bernardeau et al. (2003)
and applied to a cosmic shear survey in Bernardeau et al. (2002) 
it is a particularly convenient one owing to its simple relation to
the bispectrum.

The aperture measures are the easiest way to separate E and Bmodes
of the shear field. Essentially all E/Bmode decompositions for the
secondorder shear statistics have been performed using the aperture
measures, and we expect that they will play the same role for the
thirdorder statistics. Furthermore, since two of the four independent
combinations (73) of the aperture measures are expected
to vanish because of parity invariance, they provide a very convenient way
to detect remaining systematics in the observing, data reduction and
analysis process.
 The aperture statistics are also easily obtained from numerical
raytracing simulations, since they are defined in terms of the
underlying convergence in the first place. Hence, in these simulations
one can work directly in terms of the convergence instead of the more
complicated (due to the various components) shear field.
From the derivation of the 3PCF as a function of the bispectrum, it
becomes clear that the definition of the natural components have eased
the algebra considerably, compared to the case in which one would have
tried to calculate its individual components (like
). Furthermore, the derivation of the thirdorder
aperture statistics directly requires the combination of the shear
3PCF provided by the natural components. As discussed in Paper I there
are various ways to define the natural components of the shear 3PCF,
corresponding to the different centers of a triangle. The derivation
of the aperture measures in terms of the 3PCF has yielded the result
that the projection with respect to the centerofmass of a triangle
is the most convenient definition (at least in this connection).
Acknowledgements
We like to thank Mike Jarvis and Lindsay King for helpful comments on
this manuscript. In particular, we are grateful to Mike Jarvis for
pointing out quite a number of typos and errors in the previous
version. We also thank the anonymous referee who, by giving us
a bit of a hard time, has forced us to make some points
considerably clearer than they were in the original version and
thus has helped us to improve the paper, not least by asking to
add two figures.
This work was supported by the German Ministry for
Science and Education (BMBF) through the DLR under the project 50 OR
0106, and by the Deutsche Forschungsgemeinschaft under the project
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Copyright ESO 2005