A&A 389, 729-741 (2002)
DOI: 10.1051/0004-6361:20020626
P. Schneider ^{1,2} - L. van Waerbeke^{3,4} - Y. Mellier^{3,5}
1 - Institut f. Astrophysik u. Extr. Forschung, Universität Bonn,
Auf dem Hügel 71, 53121 Bonn, Germany
2 -
Max-Planck-Institut f. Astrophysik, Postfach 1317,
85741 Garching, Germany
3 -
Institute d'Astrophysique de Paris, 98 bis, boulevard
Arago, 75014 Paris, France
4 -
Canadian Institute for Theoretical Astrophysics, 60 St
Georges Str., Toronto, M5S 3H8 Ontario, Canada
5 -
Observatoire de Paris, DEMIRM/LERMA, 61 avenue de
l'Observatoire, 75014 Paris, France
Received 19 December 2001 / Accepted 19 April 2002
Abstract
Weak gravitational lensing by the large scale
structure can be used to probe the dark matter distribution in
the Universe directly and thus to probe cosmological
models. The recent detection of cosmic shear by several groups
has demonstrated the feasibility of this new mode of
observational cosmology. In the currently most extensive
analysis of cosmic shear, it was found that the shear field
contains unexpected modes, so-called B-modes, which are
thought to be unaccountable for by lensing. B-modes can in
principle be generated by an intrinsic alignment of galaxies
from which the shear is measured, or may signify some
remaining systematics in the data reduction and analysis. In
this paper we show that B-modes in fact are produced by
lensing itself. The effect comes about through the clustering
of source galaxies, which in particular implies an angular
separation-dependent clustering in redshift. After presenting
the theory of the decomposition of a general shear field
into E- and B-modes, we calculate their respective power
spectra and correlation functions for a clustered source
distribution. Numerical and analytical estimates of the
relative strength of these two modes show that the resulting
B-mode is very small on angular scales larger than a few
arcminutes, but its relative contribution rises quickly
towards smaller angular scales, with comparable power in both
modes at a few arcseconds. The relevance of this effect with
regard to the current cosmic shear surveys is discussed; it
can not account for the apparent detection of a B-mode
contribution on large angular scales in the cosmic shear
analysis of van Waerbeke et al. (2002).
Key words: cosmology - gravitational lensing - large-scale structure of the Universe
It was only in 2000 when four teams nearly simultaneously and independently announced the first detections of cosmic shear from wide-field imaging data (Bacon et al. 2000; Kaiser et al. 2000; van Waerbeke et al. 2000; Wittman et al. 2000). The detections reported in these papers (and in Maoli et al. 2001, using the VLT, and Rhodes et al. 2001, using HST images obtained with the WFPC2 camera) concerned various two-point statistics, like the shear dispersion in an aperture, or the shear correlation function. In van Waerbeke et al. (2001), the aforementioned statistics, as well as the aperture mass statistics (SvWJK), were inferred from the effective 6.5 square degrees of high-quality imaging data. Very recently, Hämmerle et al. (2002) reported on a cosmic shear detection using HST parallel images taken with the STIS instrument on an effective angular scale of 30''.
The shear field, originating from the inhomogeneous matter
distribution, is a two-dimensional quantity, whereas the projected
density field of the matter is a scalar field. The relation between
the shear
and the projected matter density
is
Pen et al. (2002) pointed out that the cosmic shear data of van Waerbeke et al. (2001) contains not only an E-mode, but also a statistically significant B-mode contribution in addition. Such B-modes can be generated by effects unrelated to gravitational lensing, such as intrinsic alignment of galaxies (e.g., Heavens et al. 2000; Crittenden et al. 2001a; Croft & Metzler 2000; Catelan et al. 2000) or remaining systematics in the data reduction and analysis.
In this paper we show that a B-mode contribution to the cosmic shear is obtained by lensing itself. A B-mode is generated owing to the clustering properties of the faint galaxies from which the shear is measured. This spatial clustering implies an angular separation-dependent clustering in redshift, which is the origin not only of the B-mode of the shear, but also of an additional E-mode contribution.
The paper is organized as follows: in Sect. 2 we provide a tutorial description of the E/B-mode decomposition of a shear field. Most of the results there were derived before in Crittenden et al. (2001b, hereafter C01), but we formulate them in standard lensing notation, which will be needed for the later investigation. The calculation of two-point cosmic shear statistics in the presence of source clustering is presented in Sect. 3 where it is shown that this clustering produces a B-mode. Numerical and analytical estimates of the amplitude of this B-mode are provided in Sect. 4 and discussed in Sect. 5.
In this section we provide the basic relations for the decomposition of the shear field into E- and B-modes. Most of these relations have been obtained in C01; we shall write them here in standard lensing notation.
If the shear field is obtained from a projected surface mass density
as in Eq. (1), then the gradient of the density
field
is related to the first spatial derivatives of the
shear components in the following way (Kaiser 1995):
If the shear field cannot be ascribed to a single geometrically thin
gravitational lens, the non-gradient part of
is not
necessarily due to noise. For example, if the galaxies have intrinsic
alignments, this may induce a curl-part of .
To project out the
gradient and curl part of ,
we take a further derivative
of ,
and define
(4) |
An alternative way to define
and
is
through the Kaiser & Squires (1993) mass-reconstruction relation
To simplify notation and calculations, it is convenient to express
two-component quantities in terms of complex numbers. We define the E-
and B-mode potentials
and
by
(6) |
(7) |
The discussion above dealt with the shear field itself. In the application to cosmic shear, one usually does not investigate the shear of a -field itself, but its statistical properties. In this paper we shall concentrate solely on two-point statistical measures of the cosmic shear, and their decomposition into E- and B-modes.
Owing to statistical homogeneity and isotropy of the Universe,
are homogeneous and isotropic random
fields. Hence, in terms of their Fourier transforms
(8) |
= | |||
= | |||
= | (9) |
(10) |
(12) |
(13) |
We define the four correlation functions
= | |||
= | (23) |
Using the complex number
= | |||
= |
The aperture measures can be obtained directly from the observational
data by laying down a grid of points, at each of which
and
are calculated from (28). However, obtaining
the dispersion with this strategy turns out to be difficult in
practice, since data fields usually contain holes and gaps,
e.g. because of masking (for bright stars), bad columns etc. It is
therefore interesting to calculate these dispersions directly in terms
of the correlation functions, which can be done by inserting (19) into (32),
T_{-}(x) = | |
T_{+}(x) = |
Figure 1: The four functions defined in text. | |
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Another cosmic shear statistics often employed is the shear dispersion
in a circle of angular radius .
It is related to the power
spectra by
(37) |
As before, the shear dispersion can be obtained by calculating the
mean shear in circles which are laid down on a grid of points, with
the drawback of being affected by gaps in the data
field. Alternatively, the shear dispersion can be obtained directly
from the correlation function,
(39) |
One can also define the shear dispersions of the E- and B-mode,
according to
S_{-}(x) | = | ||
= | (42) |
In the previous section we have presented the decomposition of a general shear field into E/B-modes. It is usually assumed that lensing alone yields a pure E-mode shear field, so that the detection of a B-mode in the van Waerbeke et al. (2001) data (see also Pen et al. 2002) was surprising and interpreted as being due to systematic errors or a signature of intrinsic alignment of sources. Here we show that lensing indeed does generate a B-mode component of the shear if the source galaxies from which the shear is measured are clustered.
Define the "equivalent'' surface mass density for a fixed source
redshift, or comoving distance w,
(44) |
= |
The operators
only act on the final term of (47) which
can be evaluated using the Fourier transform of ,
as in (11),
(47) |
When measuring cosmic shear from source ellipticities, the source
galaxies have a broad distribution in redshift, unless information on
the redshifts are available and taken into account. Hence, to
calculate the observable shear correlation functions, the
foregoing expressions need to be averaged over the source redshift
distribution. Let
be the probability density for
comoving distances of two sources separated by an angle
on
the sky; then we have for the observable correlation functions
(53) |
(54) |
One can check that the correlated redshift probability distribution
behaves as expected in some simple cases. For example, if
is very much smaller than the characteristic source distance w_{0},
one finds that
We can now rewrite (50) in the form
From the correlation functions (56), by writing
= | |||
(58) |
(59) |
Using the definitions of the E- and B-mode correlation function, we
obtain
(61) | |||
(62) | |||
(63) | |||
(64) |
Whereas the presence of a B-mode, and an additional contribution to
the E-mode due to source clustering must occur, one needs to estimate
the relative amplitude of this effect as compared to the "usual''
cosmic shear strength described by .
This estimate requires
a model for the source clustering, i.e., a model for the function
.
g can be related to the three-dimensional correlation
function
of galaxies,
(65) |
(66) |
(67) |
For this reason, we shall assume the power-law dependence of
as given above; in addition, we will make the
simplifying assumption that the comoving clustering length r_{0}(w) is
independent of distance w; this assumption is not too
critical, since the function
is relatively
well peaked and therefore large w-variations of the correlation
length are not probed. Then, (69) determines this constant
comoving correlation length r_{0}. We obtain in this case
In order to make further progress, we need to assume a redshift
distribution for the sources from which the shear is measured. We
employ the form (Brainerd et al. 1996)
(69) |
Figure 2: Dimensionless power spectra , as a function of of wavenumber . The solid curve corresponds to the power spectrum that is the "standard'' power spectrum of the projected mass density. The dotted curve displays , and the two dashed curves correspond to the E- and B-mode power caused by the source clustering. Here, a CDM model was used, with shape parameter , normalization , and the source redshift distribution is characterized by z_{0}=1, yielding . Other parameters for the model used here are mentioned in the text. | |
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In Fig. 2 we show an example of the various power spectra considered here; all power spectra are multiplied by to obtain dimensionless quantities. For this figure, we employed a standard CDM model with and and normalization . Sources are distributed in redshift according to the foregoing prescription, with z_{0}=1. The amplitude of the angular correlation function of galaxies was chosen to be A(1')=0.02, and the slope of for the three-dimensional correlation function was used; the corresponding correlation length in this case is . To calculate the three-dimensional power spectrum and its redshift evolution, we used the Peacock & Dodds (1996) prescription for the non-linear evolution of .
The power spectrum of the projected mass density, , is the same as that in SvWJK, except for a slightly different choice of the cosmological parameters. The spectrum is very much smaller than , as expected from the smallness of the amplitude of the angular correlation function; in fact, the ratio is nearly constant at a value of approximately (1+B)A(1'), with for this choice of the parameters.
The behavior of the power spectra which arise from source clustering, and , as a function of is quite different. First, both of these spectra are very similar, which is due to the fact that the J_{4}-term in (61) is much smaller than the J_{0}-term. Second, although both of these spectra are small on large angular scales, i.e. at small , their relative value increases strongly for smaller angular scales. Hence, as expected, the relative importance of source clustering increases for larger . What is surprising, though, is that these power spectra have the same amplitude as at a value of , corresponding to an angular scale of , and the relative contribution of the B-mode amounts to about 2% at an angular scale of 1'. It should be noted here that cosmic shear has already been measured on scales below 1'; therefore, source clustering gives rise to a B-mode component in cosmic shear which is observable.
We shall now consider the behavior of
for large
values of .
The aforementioned properties of
can be summarized as
Figure 3: For the same model as in Fig. 2, several correlation functions are plotted. The solid line shows ; in fact, the correlation function cannot be distinguished from on the scale of this figure; their fractional difference is less than 1%, even on the smallest scale shown. The two B-mode correlation functions are shown as well as and . Note that the difference between the latter two is larger than that of the corresponding "+''-correlation functions. | |
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In Fig. 3 we have plotted several correlation functions; they have been calculated from the power spectra plotted in Fig. 2 by using (20). The first point to note is that differs from by less than 1% for angular scales larger than 3''; hence, the relative contribution caused by the source correlation is even smaller than that seen in the power spectra. This is due to the fact that the correlation function is a filtered version of the power spectra, however with a very broad filter. This implies that even at small scales, the correlation function is not dominated by large values of , where the contribution from source clustering is largest, but low values of contribute significantly. The influence of source clustering on the "-'' modes is larger, since the filtering function for those are narrower (i.e., J_{4}(x) is a more localized function that J_{0}(x)), and differs from appreciably on scales below about 1'.
Finally, in Fig. 4 we have plotted the aperture measures. On scales
below about 1', the dispersion of
is larger than about 1% of
that of
.
Hence, the ratio of these E- and B-mode aperture
measures are very similar to that of the corresponding power spectra.
Figure 4: Aperture measures, for the same model as used in Fig. 2. Shown here is the dispersion of the aperture mass, , the corresponding function in the absence of source correlations (noted by the subscript "0'') and , which is the aperture measure for the B-mode. As expected from the power spectra shown in Fig. 2, and the fact that the aperture measures are a filtered version of the power spectra with a very narrow filter function, the B-mode aperture measure is considerably smaller than itself. | |
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The fact that and are very similar in amplitude means that by measuring , one can make an approximate correction of , obtaining a value close to by subtracting from . Owing to the relative amplitude of these correlation-induced powers, such a correction may be needed in future high-precision measurements of the cosmic shear.
We have shown that the clustering of galaxies from which the shear is measured leads to the presence of a B-mode in the cosmic shear field, in addition to providing an additional component to the E-mode. The reason for this effect in essence is the angular separation-dependent redshift correlation of galaxies, which causes the mean of the product of the angular-diameter distance ratio along two lines-of-sight not to factorize, but to depend on . For a fiducial model considered in detail, the B-mode contribution amounts to more than on angular scales below 1' (or ), and its relative importance quickly rises towards smaller angles. On substantially larger angular scales, however, the B-mode contribution is small. Furthermore, the additional E-mode contribution is very similar in size to the B-mode power, which will allow an approximate correction of the measured E-mode for this additional term.
From an observational point-of-view, the most easily accessible quantities are the shear correlation functions , as one can easily deal with gaps in the data field. In Sect. 2 we have given explicit relations regarding how other two-point statistics of the shear can be calculated in terms of the shear correlation function. The finite support of the functions indicates that the aperture measures are more easily obtained from observational data than either the E- and B-mode correlation functions, or the E- and B-mode shear dispersions. Therefore, the aperture measures are the preferred method to check for the presence of a B-mode contribution in the shear data.
We have varied some of the model parameters; in particular, we have considered the case of lower mean source redshift (corresponding to a brighter flux threshold), and simultaneously increasing A(1'), such that the clustering length r_{0} stays about the same. In this case we found a very similar ratio between the B- and E-mode power spectra as for the example considered in Sect. 4. We consider it unlikely that the observed B-mode in the present day data sets is due to the source clustering effect. The B-mode found in van Waerbeke et al.(2001) and Pen et al. (2002) can actually be used to search quantitatively for residual systematics. Its detection in van Waerbeke et al.(2001) was done by obtaining and by laying down a grid of circular apertures on the data field. A more accurate measurement of and has been obtained from the same data by Pen et al. (2002), by calculating them from the observed correlation functions , as in (33). In fact a subsequent analysis revealed that the B-mode measured in these data were essentially residual systematics caused by an overcorrection of the PSF, and can be corrected for (van Waerbeke et al. 2002). In this latter analysis, no significant B-modes are detected at small angular scales, but on scales above 10', slightly significant values of are detected; the effect discussed in this paper can certainly not account for them.
The effect considered here seems to have been overlooked hitherto. Bernardeau (1998) considered the effects of source clustering on cosmic shear statistics and concluded that this source clustering can strongly affect the skewness and kurtosis of the cosmic shear, but to first order leaves the shear dispersion (and thus the power spectrum) unaffected. Hamana et al. (2002) studied this effect with ray tracing simulations, again concentrating on the skewness. Most of the other ray tracing simulations of weak lensing (e.g., van Waerbeke et al. 1999; Jain et al. 2000) assumed all sources to be at the same redshift, in which case the additional power discussed here does not occur. Lombardi et al. (2002) calculated the effect of source clustering on the noise of weak lensing mass maps, showing that it can provide a significant noise contribution in the inner regions of clusters.
It must be pointed out that the effect considered here is unrelated to other lensing effects which in principle could generate a B-mode, such as lens-lens coupling or the break-down of the Born approximation (see Bernardeau et al. 1997 and SvWJK for a discussion of these two effects on the skewness). Numerical estimates (e.g., Jain et al. 2000) show that these latter two effects are very weak. Bertin & Lombardi (2001) considered the situation of lensing by two mass concentrations along the line-of-sight, where a B-mode is generated by a strong lens-lens coupling, but the fraction of lines-of-sight where this occurs is tiny. Another effect which could in principle generate a B-mode from lensing is the fact that the observable is not the shear itself, but the reduced shear (Schneider & Seitz 1995). In the appendix we show that this effect is indeed negligible.
Like the intrinsic alignment of galaxies, which can yield a spurious contribution to the measured cosmic shear, the source clustering effect can in principle be avoided if redshift estimates of the source galaxies are available. In that case, by estimating the shear correlation function, pairs of galaxies with a large likelihood to be at the same distance can be neglected. In contrast to the intrinsic correlation of galaxies, the B-mode from source clustering appears to be fairly insensitive to the redshift distribution of the source galaxies, provided the clustering length is kept fixed.
Acknowledgements
We thank L. J. King for useful comments on the manuscript. This work was supported by the TMR Network "Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Matter'' of the EC under contract No. ERBFMRX-CT97-0172 and by the German Ministry for Science and Education (BMBF) through the DLR under the project 50 OR 0106.
The shear is not directly an observable, but is estimated from the image ellipticities of distant galaxies. The expectation value of the image ellipticity, however, is not the shear, but the reduced shear . Hence, the correlation of the observed ellipticities is the correlation of the reduced shear, not the shear itself. In cosmic shear, nearly everywhere, and so the difference between shear and reduced shear shall not play a big role. However, at least a priori, this effect cannot be neglected, as seen from the following argument:
The skewness , where X is a measure of shear (such as , or the reconstructed ) has been calculated by van Waerbeke et al. (2001) to be of order a few hundred. On a scale of about one arcminute, , so that , taking for the top-hat smoothed . The difference between the correlation functions involving g and those involving is in principle of the same order-of-magnitude as and thus can be present at the level of a few percent, and there is no reason why it should not contain a B-mode contribution.
We define the correlation functions
(A.1) |
(A.2) |
(A.3) |
(A.5) |
= | |
= | (A.7) |
(A.8) |
(A.9) |
(A.10) |