## 3. The Weak-Lensing equations## 3.1. Description of the deformation in the image planeThe 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 where in the case of an homogeneous Universe,
, 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 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), In particular the local convergence is given by the trace of this matrix, through the equation, 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)
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 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 matrixWe 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 symmetric tidal matrix, and depends on the second derivatives of the Newtonian gravitational potential (Sachs 1961, 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 . Let be the 3-dimensional Newtonian gravitational potential. The comoving Poisson equation is, 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, From Eqs. (7) and (16) the central equation driving the matrix is then, 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 Since , for a completely homogeneous universe, the matrix is given by, and we are left with the equation for the distance We know that the solution of this differential equation is given by with 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 convergenceIn this section we restrict ourself to the 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, with To solve this differential equation it is more convenient to write
it with the variable 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 where is the distance between the redshifts
and 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 functionWe call in Eq. (43) the efficiency
function. It describes the efficiency with which a lens at a given
redshift 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 |