**B.4.2. The Effective Single Screen Lens**

Throughout these notes we will treat lenses as if all the lens components lay at a single redshift ("the single screen approximation"). The lens equations for handling multiple deflection screens (e.g. Blandford & Narayan [1986], Kovner [1987b], Barkana [1996]) are known but little used except for numerical studies (e.g. Kochanek & Apostolakis [1998], Moller & Blain [2001]) in large part because few lenses require multiple lens galaxies at different redshifts with the exception of B2114+022 (Chae, Mao & Augusto [2001]). In fact, we are not being as cavalier in making this approximation as it may seem.

The vast majority of strong lenses consist of a single lens galaxy perturbed by other objects. We can divide these objects into those near the primary lens, where a single screen is clearly appropriate, and those distributed along the line of site for which a single screen may be inappropriate. Because the correlation function is so strong on small scales, the perturbations are dominated by objects within a correlation length of the lens galaxy (e.g. Keeton, Kochanek & Seljak [1997], Holder & Schechter [2003]). The key to the relative safety of the single screen model is that weak perturbations from objects along the line of site, in the sense that in a multi-screen lens model they could be treated as a convergence and a shear, can be reduced to a single "effective" lens plane in which the true amplitudes of the convergence and shear are rescaled by distance ratios to convert them from their true redshifts to the redshift of the single screen (Kovner [1987b], Barkana [1996]). The lens equation on the effective single screen takes the form

(B.63) |

where *F*_{OS}, *F*_{LS} and
*F*_{OL} describe the shear and convergence
due to perturbations between the observer and the source, the lens and
the source and the observer and the lens respectively. For statistical
calculations this can be simplified still further by making the coordinate
transformation
' =
(*I* + *F*_{OL})
and
'
= (*I* + *F*_{LS})
)
to leave a lens equation,

(B.64) |

identical to a single screen lens
in an effective convergence and shear of
*F*_{e} = *F*_{OL} + *F*_{LS} -
*F*_{OS} (to linear
order). In practice it will usually be safe to neglect the differences
between Eqns. B.63 and B.64 because the shearing terms
affecting the deflections in Eqn B.63 are easily mimicked by
modest changes in the ellipticity and orientation of the primary lens.
The rms amplitudes of these perturbations depend on the cosmological model
and the amplitude of the non-linear power spectrum, but the general scaling
is that the perturbations grow as *D*_{s}^{3/2}
with source redshift, and increase for larger
_{8} and
_{M} as
shown in Fig. B.19 from Keeton et al.
([1997]).
The importance of these
effects is very similar to concerns about the effects of lenses along the
line of sight on the brightness of high redshift supernova being used to
estimate the cosmological model (e.g. Dalal et al.
[2003]).