3. The Weak-Lensing equations
3.1. Description of the deformation in the image plane
The aim of this subsection is to present the mathematical objects that describe the local gravitational deformation. In particular we recall the relationships between the local convergence , the statistical properties of which we will investigate, and the quantities that are more directly observable. We consider two neighboring geodesics and which are enclosed by a light bundle, and we call the two dimensional position angle between and at the observer position. Let us define as the angular diameter distance between and at a given redshift z. We assume that the small angle approximation is valid, and we have,
where in the case of an homogeneous Universe, , a being the expansion factor, the Kronecker symbol, and the physical distance, (see definition hereafter). In an inhomogeneous Universe describes the deformation of the light bundle produced by the matter distribution. The angular distance corresponds to an angular position on the sky such that,
The shapes of the light bundles on the image plane are described by the amplification matrix , the Jacobian of the transformation from a virtual screen located at z (the source plane), to the image plane as seen by the observer. Its inverse, , is actually more closely related to observable quantities,
The amplification matrix can be expressed in terms of the geometrical deformation of the light bundle, the convergence and the shear components and (Schneider et al. 1992),
The intensity of the shear is given by the ratio of the eigenvalues of the matrix . The relations between the physical quantities and the observable quantities, the magnification (=amplification) µ and the distortion of the images, are given by,
where . Note that in the weak lensing regime we have the simple relations,
In practice, the magnification is measured through the local density of objects (Broadhurst et al. 1995, Broadhurst 1995, Fort et al. 1996) or through the image size of galaxies in different surface brightness slices (Bartelmann & Narayan 1995), whereas the distortion is measured directly from the shape of the background galaxies (Bonnet al. 1994, Fahlman et al. 1994, Smail et al. 1994, Squires et al. 1996a, b). One should be careful when order moments of the local convergence higher than the variance are considered, since non-linear couplings exist in general between µ and , and Eqs. (13) cannot be used. It is possible to get the convergence from Eqs. (12) by the measurement of both the magnification and the distortion, or by using only the distortion and the relation introduced by Kaiser (1995, see also Seitz & Schneider 1996) which is valid even beyond the linear regime,
In the following we will thus assume that the local convergence is accessible to the observations in a given sample, and we will focus our investigations on the statistical properties of this quantity.
3.2. The source equation for the deformation matrix
We perform the calculations assuming that the Born approximation is valid, i.e. that the deformation of a light bundle can be calculated along the unperturbed geodesics. This assumption will be discussed in Sect. 5.7. In a Friedmann-Robertson-Walker (FRW) Universe, such a geodesic may be parameterized by the time variable , defined by,
where is the (time dependent) Hubble constant (in units of ). Using the Born approximation and the geodesic deviation equation, the evolution equation for the angular diameter distance in a lumpy Universe, along the unperturbed ray, as a function of is (Seitz 1993),
where is the Ricci tensor, is the Weyl tensor, is the wave-vector of the light ray and is a complex null vector propagated along the geodesic such that .
where is the density contrast, is the 3-dimensional Laplacian operator, and the derivatives are done with respect to the proper distance. Note that Eq. (20) does not depend on the cosmological constant . Then, it is straightforward to show that,
with the boundary conditions,
The first equation expresses the focusing condition at the observer, and the second the Euclidean properties of the space at small redshift. is a symmetric matrix that can be written in terms of , and using the expressions of R and F,
Since , for a completely homogeneous universe, the matrix is given by,
We know that the solution of this differential equation is given by
This equation has a known analytical solution only when the cosmological constant is zero. Otherwise it has to be integrated numerically.
3.3. The expression of the local convergence
In this section we restrict ourself to the linear regime. We then assume that the difference between the local value of and its value for a homogeneous universe is small, so that,
where we have introduced an effective potential such that,
where is the local mass over-density in the linear regime. We assume the small angle deviation approximation is valid. In that case, the plane-parallel approximation works, which means that only waves perpendicular to the line-of-sight contribute to lensing, and consequently we can neglect the second order derivative along the unperturbed ray in the Laplacian operator. Moreover, the difference is a small quantity, which can be expanded with respect to the initial density. We then define as the part of this difference which is linear in this density field (terms of higher order will be considered in Sect. 5). The differential equation for can then be derived from (23) and it reads,
To solve this differential equation it is more convenient to write it with the variable z. The differential equation then reads,
The homogeneous differential equation associated with this equation has two known solutions, the distance functions and are given respectively by Eq. (28) and,
from which the solution of the differential equation (33) can be written. Indeed the solution of the inhomogeneous differential equation reads
which, after elementary mathematical transformations simplifies in,
It follows from Eq. (11) that the local convergence can be written for a source at redshift z (see also Seljak 1996),
where is the distance between the redshifts and z. We have also introduced the new variable , very useful for such calculations, which is the physical distance along the line-of-sight,
It coincides with only for a flat geometry . Eq. (40) expresses the fact that the local convergence is given by the superposition of the convergence induced by each lens between the observer and the source. For the distribution of sources we thus have
The previous two equations are the basic ones that relate the local convergence to the linear cosmic density along the line-of-sight for any cosmological model.
3.4. The efficiency function
We call in Eq. (43) the efficiency function. It describes the efficiency with which a lens at a given redshift z located along the line-of-sight will affect the local convergence. This efficiency function obviously depends on the redshift of the sources. It is maximum at half the distance between the source and the observer, and vanishes at both ends. In Fig. 1, we present the shape of the efficiency function in different case. In 1a, this is for an Einstein-de Sitter Universe for two different populations of sources. The thin lines show the shape of the distribution functions of the redshifts of the sources that are assumed to be either centered on (solid line) or (dashed line). The typical redshift of the lens is about 0.4 to 0.5, with a broader distribution when the redshift of the sources is larger.
In the lower panel we show the dependence of the efficiency function on the cosmological parameters. Here the sources are simply assumed to all lie at . We can see that the shape of the efficiency function is not really affected. The amplitude is however changed, and it is roughly proportional to . The dependence on , although not totally negligible, is considerably weaker than that on .
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998